Groups, graphs, and hypergraphs: average sizes of kernels of generic matrices with support constraints
GGroups, graphs, and hypergraphs: averagesizes of kernels of generic matrices with support constraints
Tobias Rossmann and Christopher Voll
We develop a theory of average sizes of kernels of generic matrices with supportconstraints defined in terms of graphs and hypergraphs. We apply this theory tostudy unipotent groups associated with graphs. In particular, we establish stronguniformity results pertaining to zeta functions enumerating conjugacy classes of thesegroups. We deduce that the numbers of conjugacy classes of F q -points of the groupsunder consideration depend polynomially on q . Our approach combines group theory,graph theory, toric geometry, and p -adic integration.Our uniformity results are in line with a conjecture of Higman on the numbersof conjugacy classes of unitriangular matrix groups. Our findings are, however, instark contrast to related results by Belkale and Brosnan on the numbers of genericsymmetric matrices of given rank associated with graphs. Contents
Keywords.
Unipotent groups, p -groups, conjugacy classes, graphs, hypergraphs, average sizes ofkernels, zeta functions, p -adic integration, toric geometry, generic matrices, cographs, weak orders a r X i v : . [ m a t h . G R ] A ug Introduction
In this article, we study enumerative questions related to spaces of matrices defined viasupport constraints. Our work is motivated by and has immediate applications to thestudy of (conjugacy) class numbers of finite p -groups. We will naturally touch threesubjects: rank distributions in spaces of matrices, class numbers of unipotent groups, andzeta functions of groups. We begin by summarising key facts from each of these fields. Polynomiality.
The study of rank distributions in combinatorially defined spaces ofmatrices has a long history and draws on contributions from several fields of mathematicalresearch. The numbers of arbitrary [42] n × m matrices or of antisymmetric [16, Theorem 3],symmetric [49, Theorem 2], or traceless [7, 15] n × n matrices of a given rank over afinite field F q are each given by an explicitly known polynomial in q ; we assume that q odd in the (anti-)symmetric cases. Lewis et al. [44] and Klein et al. [39] obtainedfurther polynomiality results for rank distributions in spaces of general, symmetric, andantisymmetric matrices obtained by insisting that entries in suitable positions be zero. Wilderness.
The study of rank distributions naturally involves algebro-geometric meth-ods. Thanks to these, much is known about ideals of minors associated with generic,symmetric, and antisymmetric matrices [14, 71].In drastic contrast to the polynomiality results above, Belkale and Brosnan [6, Theo-rem 0.5] demonstrated that enumerating matrices of a given rank is a “wild” problem,even for spaces of combinatorial origin. More precisely, given n (cid:62) S , con-sider the space Sym n ( F q ; S ) of symmetric n × n matrices [ a ij ] over F q with a ij = 0whenever ( i, j ) S . Belkale and Brosnan showed that, in a precise technical sense,enumerating invertible matrices in Sym n ( F q ; S ) is as difficult as counting F q -points onarbitrary Z -defined varieties. (To the authors’ knowledge, it is unknown whether thesame conclusion holds for spaces of arbitrary or antisymmetric matrices with suitablyconstrained supports.) Belkale and Brosnan used their result to refute a conjecture ofKontsevich on the polynomiality of the numbers of F q -points of specific hypersurfacesassociated with graphs.Halasi and Pálfy [34] obtained results of a similar flavour on the numbers of matrices overfinite fields that satisfy prescribed “rank restraints”. In this context, too, polynomialityresults (requiring fairly restrictive combinatorial assumptions) mark the exception fromthe rule of “wild” variation of the relevant numbers with the prime power q . Class numbers.
Let k( G ) denote number of conjugacy classes (“class number”) of afinite group G . Let U n ( R ) be the group of upper unitriangular n × n matrices over aring R . In an influential paper [36], Higman asked whether k(U n ( F q )) is always given2 Introduction by a polynomial in q . This question has been answered affirmatively for n (cid:54)
13 byVera-López and Arregi [70] and for n (cid:54)
16 by Pak and Soffer [53].
Beyond Higman’s conjecture.
We are interested in problems in the spirit of Higman’squestion for other types of unipotent groups. Let G (cid:54) U n be a subgroup scheme—wemay think of G as a subgroup of U n ( C ) defined by the vanishing of polynomials withinteger coefficients. What can be said about the class numbers k( G ( F q )) as a functionof q ? In particular, when does k( G ( F q )) depend polynomially on q ?Questions like these have been asked and answered, to varying degrees of generality, fornumerous group schemes realising e.g. (Sylow subgroups of) Chevalley groups or relativelyfree p -groups of exponent p , by numerous authors, including the above-mentioned andEvseev, Goodwin, Isaacs, Le, Lehrer, Magaard, Mozgovoy, Röhrle, and Robinson, toname but a few; see, for instance, [29, 32, 51, 52] and the references therein.The aforementioned problems are closely related to the enumeration of matrices ofgiven rank in Z -defined spaces of matrices over F q . We note, for example, that thework in [34] on matrices with given rank constraints was motivated by a study of classnumbers of pattern groups, viz. certain combinatorially defined subgroups of U n ( F q ).This connection also occurs in previous work [52, 57, 60] of both authors. Moreover, itturns out that if we are willing to exclude small exceptional characteristics, the studyof k( G ( F q )) for group schemes G (cid:54) U n as above essentially reduces to those of class 2.(For a proof, combine [60, Proposition 6.4] and [60, Lemma 7.1].) Alternating bilinear maps.
As a variation of the classical Baer correspondence [2], wemay construct a (unipotent) group scheme G (cid:5) of class at most 2 from each alternatingbilinear map (cid:5) : Z n × Z n → M , where M is a free Z -module of finite rank; see §2.4 fordetails. We call G (cid:5) the Baer group scheme associated with (cid:5) . Commutators in G (cid:5) are given by (cid:5) . For example, if (cid:5) : Z × Z → Z is the standard symplectic form, then theassociated Baer group scheme is U , the Heisenberg group scheme.Rather than consider the maps (cid:5) , we may equivalently use antisymmetric matrices.Let so n ( Z ) ⊂ M n ( Z ) denote the module of antisymmetric n × n matrices over Z . Thenevery Z -module homomorphism M θ −→ so n ( Z ) defines an alternating bilinear map[ θ ]: Z n × Z n → M ∗ := Hom( M, Z )such that x [ θ ] y is the functional a x ( aθ ) y > . In particular, for a submodule M ⊂ so n ( Z ),we obtain a Baer group scheme G M := G [ ι ] , where ι is the inclusion M , → so n ( Z ).We note that using essentially a variation of the above construction, finite p -groupsassociated with spaces of antisymmetric matrices over F p have found applications relatinggraph- and group-theoretic problems in recent work of Bei et al. [5] and Li and Qiao [45]. Average sizes of kernels.
Again, up to excluding small characteristics, as a functionof q , the study of the class numbers k( G M ( F q )) for modules M ⊂ so n ( Z ) turns out to beessentially equivalent to the study of k( G ( F q )) for arbitrary unipotent group schemes G .By focusing on the Baer group schemes of the form G M , we may easily relate the study3 Introduction of class numbers to that of enumerating antisymmetric matrices of given rank as in §1.1.Namely, if A is a finite ring and if ¯ M ⊂ so n ( A ) denotes the submodule generated by theimage of M , then k( G M ( A )) = | A | m · | ¯ M | X a ∈ ¯ M | Ker(¯ a ) | , where m is the rank of M as a Z -module; cf. Proposition 2.5. That is, up to a harmlessfactor, the class number of G M ( A ) is the average size of the kernels of the elements of ¯ M acting on A n . Thus, if A = F q is a finite field, then we may express k( G M ( F q )) in termsof the numbers of F q -points of the Z -defined rank loci in M ; see [57, §2.1] for details. We have seen that the study of the class numbers k( G ( F q )) for group schemes G (cid:54) U n is intimately related to the study of average sizes of kernels of matrices over F q . Classcounting and ask zeta functions provide convenient tools for generalising this connection tomuch more general finite rings, including those of form Z /p k Z ( p prime) and F q [ z ] / ( z k ). Class counting zeta functions.
Let R be the ring of integers of a local or a global field.Let G be a group scheme of finite type over R . The class counting zeta function of G is the Dirichlet series ζ cc G ( s ) := X = I/R k( G ( R/I )) · | R/I | − s . Class counting zeta functions were introduced by du Sautoy [23] for p -adic linear groups.They were further studied by Berman et al. [8] for Chevalley groups and by the firstauthor [57, 60] and Lins [46–48] for unipotent groups; other names for these functions inthe literature are “conjugacy class zeta functions” and “class number zeta functions”. Theuse of zeta functions as a tool in group theory was pioneered by Grunewald et al. [33]. Euler products and variation of the place.
As we will now explain, the study of classcounting zeta functions in characteristic zero immediately reduces to a local analysis.Let K be a number field with ring of integers O . Let V K be the set of non-Archimedeanplaces of K . For v ∈ V K , let O v denote the valuation ring of the v -adic completion of K .Let K v denote the residue field of O v and let q v = | K v | . Let G be a group scheme of finitetype over O . Then the Chinese remainder theorem yields an Euler product factorisation ζ cc G ( s ) = Y v ∈V K ζ cc G ⊗O v ( s ) . (1.1)For a general G , it is unknown how the Euler factors ζ cc G ⊗O v ( s ) vary with the place v .However, if G is a Chevalley group [8] or unipotent [57], then ζ cc G ⊗O v ( s ) can, for almostall v ∈ V K , be expressed in terms of the numbers of K v -points of certain O -definedvarieties and rational functions in q v and q − sv . In both cases, it is an open problem toprove meaningful theorems on the class of varieties “required” to describe ζ cc G ⊗O v ( s ).4 Introduction
Uniformity.
Among the ways that the Euler factors of a class counting zeta function asabove might depend on the place, the tamest conceivable case has played a central rolein the literature. Namely, we say that the group scheme G over the ring of integers O ofa number field K has uniform class counting zeta functions if there exists a rationalfunction W ( X, T ) ∈ Q ( X, T ) such that ζ cc G ⊗O v ( s ) = W ( q v , q − sv ) for all v ∈ V K . Forexample, if ζ K denotes the Dedekind zeta function of K , then ζ ccU ⊗O ( s ) = ζ K ( s − ζ K ( s − ζ K ( s ) = Y v ∈V K W ( q v , q − sv )where W ( X, T ) = − T (1 − XT )(1 − X T ) ; see [8, §8.2] and [57, §9.3]. A natural variation ofHigman’s question asks if the class counting zeta function of each U n is uniform. Whilethe above notion of uniformity is natural in view of the Euler product (1.1), both strongerand weaker concepts are frequently of interest.We wish to add a further direction by allowing local base extensions. Namely, wesay that G as above has strongly uniform class counting zeta functions if there exists W ( X, T ) ∈ Q ( X, T ) such that for all compact discrete valuation rings (
DVR s) O endowedwith an O -algebra structure, we have ζ cc G ⊗ O ( s ) = W ( q, q − s ), where q denotes the residuefield size of O . (Note, in particular, that we do not insist that O has characteristic zero.)Again, U is an example of a group scheme with strongly uniform class counting zetafunctions; this can be verified directly or deduced from much more general results below.While it is relatively easy to produce examples of group schemes with non-uniformclass counting zeta functions (see [57, §7]), as in the study of class numbers over F q in §1.2, it remains unknown just how erratically Euler factors of class counting zetafunctions may vary with the place. Ask zeta functions: analytic form.
In the same way that the class counting zetafunction ζ cc G ( s ) (and its Euler factors) of a group scheme G generalises the collection ofclass numbers k( G ( F p )) as p ranges over the primes, we may similarly define a Dirichletseries which generalises the average sizes of kernels that appeared in §1.2.Let R be a commutative ring. Consider an R -module homomorphism M θ −→ M n × m ( R ),where M is finitely generated. If R is finite, then the average size of the kernel associatedwith θ is the rational number ask( θ ) := 1 | M | X a ∈ M | Ker( aθ ) | . For each R -algebra S , we obtain a map M ⊗ S θ S −→ M n × m ( S ). Suppose that R is the ringof integers of a local or global field. The (analytic) ask zeta function [57, 60] of θ is ζ ask θ ( s ) := X = I/R ask( θ R/I ) · | R/I | − s . Introduction
Class counting and ask zeta functions.
The following by-product of [60] (to be provedin §2.4) asserts that ask zeta functions associated with modules of antisymmetric matricesessentially coincide with class counting zeta functions of associated group schemes.
Proposition 1.1.
Let M ι , −→ so n ( Z ) be the inclusion of a submodule of Z -rank m . Definea unipotent group scheme G M as in §1.2. Let R be ring of integers of a local or globalfield of arbitrary characteristic. Then ζ cc G M ⊗ R ( s ) = ζ ask ι R ( s − m ) . Beyond antisymmetric matrices, by [57, §8], ask zeta functions associated with general Z -module homomorphisms M → M n × m ( Z ) are of natural group-theoretic interest: theyenumerate linear orbits of suitable groups (although perhaps not conjugacy classes). Ask zeta functions: algebraic form.
In the local case, it will be convenient to switchfreely between the above ask zeta functions and the following algebraic counterpart.Let O be a compact DVR and let M θ −→ M n × m ( O ) be an O -linear map, where M isfinitely generated. Let P be the maximal ideal of O . Then Z ask θ ( T ) := ∞ X k =0 ask( θ O / P k ) T k ∈ Q [[ T ]]is the (algebraic) ask zeta function of θ . The Dirichlet series ζ ask θ ( s ) and ordinarygenerating function Z ask θ ( T ) determine each other in the sense that ζ ask θ ( s ) = Z ask θ ( q − s ) , where q = | O / P | denotes the residue field size of O . For this reason, we shall call each ofthese functions “the” ask zeta function of θ . In this section, we introduce (somewhat informally) the real protagonists of the presentarticle: graphical groups and adjacency and incidence representations of graphs andhypergraphs, respectively; a more complete and rigorous account will be given in §3.Throughout, graphs are finite without parallel edges but they may contain loops;graphs without loops are simple . Graphical groups and negative adjacency representations.
Let Γ be a simple graph;for simplicity, we assume that 1 , . . . , n are the vertices of Γ.The following construction of a module of antisymmetric matrices derived from Γwas used by Tutte [68]. Let e ij denote the n × n matrix with entry 1 in position ( i, j )and zeros elsewhere. Let M − (Γ) ⊂ so n ( Z ) be the submodule generated by all matrices e ij − e ji , where ( i, j ) runs over pairs of adjacent vertices. In the spirit of §1.1, M − (Γ) isthe largest module of antisymmetric n × n matrices over Z such that the support of eachmatrix in M − (Γ) is contained in the set of pairs ( i, j ) with i adjacent to j in Γ.6 Introduction
Let G Γ := G M − (Γ) be the group scheme associated with M − (Γ) as in §1.2. We call G Γ the graphical group scheme associated with Γ; see §3.4 for details. We refer to thegroups of points G Γ ( R ) over rings R as the graphical groups associated with Γ over R .For example, it is easy to see that if P n denotes the path on n vertices, then G P n ( Z ) isthe largest nilpotent quotient of class at most 2 of U n +1 ( Z ). More generally, the groups G Γ ( Z ) are precisely the class-2 quotients of right-angled Artin groups; see Remark 3.8.Among the central objects of interest in the present article are the class counting zetafunctions of graphical group schemes. Based on what we described above, the study ofthese class counting zeta functions becomes a part of the study of ask zeta functions.Namely, define the negative adjacency representation γ − of Γ to be the inclusion M − (Γ) , → so n ( Z ). By Proposition 1.1, the ask zeta function of γ − essentially coincideswith the class counting zeta function ζ cc G Γ ( s ) of the graphical group scheme G Γ . Positive adjacency representations.
As the adjective “negative” indicates, the functionsjust defined admit “positive” analogues. Suppose that Γ is a graph as before except thatwe now allow Γ to contain loops. Let M + (Γ) be the submodule of the module Sym n ( Z )of symmetric n × n matrices over Z generated by the matrices e ij + e ji for differentadjacent vertices i and j and all e ii for loops i . We define the positive adjacencyrepresentation γ + of Γ to be the inclusion M + (Γ) , → Sym n ( Z ).Even though the ask zeta functions associated with the maps γ + lack an obviousgroup-theoretic interpretation (akin to our interpretation of ζ ask γ − ( s ) in terms of the classcounting zeta function of G Γ ), they are of natural interest in light of the results dueto Belkale and Brosnan [6] mentioned in §1.1. Using our present terminology, Belkaleand Brosnan showed that, as Γ varies over all finite graphs (with loops permitted), thenumber of invertible matrices in the image of M + (Γ) ⊗ F q in Sym n ( F q ) is “arbitrarilywild” as a function of q . It is therefore natural to ask whether this wildness survivestaking the average both over F q and, similarly, on the level of suitable ask zeta functions. Hypergraphs and incidence representations.
As we saw, graphs (with loops permitted)provide a combinatorial formalism for discussing modules of antisymmetric or symmetricmatrices with support contained in a given set of positions. In the same spirit, we mayuse hypergraphs to encode modules of arbitrary rectangular matrices with constrainedsupport. Here, a hypergraph H on the vertex set { , . . . , n } consists of symbols e , . . . , e m called hyperedges and, for each j = 1 , . . . , m , a support set | e i | which is an arbitrarysubset of { , . . . , n } . Define M ( H ) ⊂ M n × m ( Z ) to be the module of all matrices [ a ij ] with a ij = 0 whenever the vertex i and hyperedge e j are not incident (i.e. whenever i
6∈ | e j | ).We refer to the inclusion M ( H ) η , −→ M n × m ( Z ) as the incidence representation of H .The ask zeta functions ζ ask η ( s ) associated with hypergraphs are of interest in view ofwork of Lewis et al. [44], Klein et al. [39], and others on rank distributions in spacesof matrices defined in terms of support constraints. In addition, over the course of thepresent article, we will encounter group-theoretic incentives for studying these functions.7 Introduction
Our first main result establishes that whatever wild geometry can be found in the rankloci of the modules M ± (Γ) ⊂ M n ( Z ) and M ( H ) ⊂ M n × m ( Z ) from §1.4 disappears onaverage in the sense that it is invisible on the level of ask zeta functions. As before, q denotes the residue field size of a compact DVR O . Theorem A (Strong uniformity) . (i) Let H be a hypergraph with incidence representation η over Z . Then there exists W H ( X, T ) ∈ Q ( X, T ) such that, for each compact DVR O , Z ask η O ( T ) = W H ( q, T ) . (ii) Let Γ be a simple graph with negative adjacency representation γ − over Z . Thenthere exists a rational function W − Γ ( X, T ) ∈ Q ( X, T ) such that, for each compact DVR O , Z ask γ O − ( T ) = W − Γ ( q, T ) . (iii) Let Γ be a (not necessarily simple) graph with positive adjacency representation γ + over Z . Then there exists a rational function W +Γ ( X, T ) ∈ Q ( X, T ) such that, foreach compact DVR O of odd residue characteristic, Z ask γ O + ( T ) = W +Γ ( q, T ) . By [57, Theorem 1.4], each of the generating functions Z ask η O ( T ), Z ask γ O − ( T ), and Z ask γ O + ( T ) inTheorem A is rational in T provided that O has characteristic zero. What is remarkableis that these functions are in fact rational in both T and q without any restrictions on O .This is not a general phenomenon for ask zeta functions; see [57, §7].The dichotomy between “tame” (i.e. strongly uniform) and “wild” behaviour is arecurring theme in the study of zeta functions associated with various group-theoreticcounting problems. Uniformity results (akin to our Theorem A) have been obtained invarious situations; see e.g. [33, Theorem 2], [66, Theorem B] and [17, Theorem 1.2].By minor abuse of notation, we refer to the rational functions W H ( X, T ) and W ± Γ ( X, T )in Theorem A as the ask zeta functions associated with H and Γ, respectively. Theserational functions are, to the best of our knowledge, new invariants of graphs andhypergraphs which, as we will see, reflect interesting structural features of the latter.An immediate consequence of Theorem A is that upon taking the average, the arbitrarilywild numbers of invertible matrices over F q provided by Belkale and Brosnan cancel. Corollary 1.2.
Let n (cid:62) and a set S be given. Define Sym n,r ( F q ; S ) to be the set ofmatrices of rank r in Sym n ( F q ; S ) (see §1.1). Then there exists a polynomial f n,S ( X ) ∈ Q [ X ] such that for each odd prime power q , n X r =0 | Sym n,r ( F q ; S ) | q n − r = f n,S ( q ) . (1.2)8 Introduction
Proof.
Let d be the F q -dimension of Sym n ( F q ; S ) and note that d does not depend on q .Let Γ be the (not necessarily simple) graph with vertices 1 , . . . , n and such that twovertices i and j of Γ are adjacent if and only if ( i, j ) , ( j, i ) ∈ S . Let f n,S ( X ) ∈ Q ( X ) bethe coefficient of T of the rational power series W +Γ ( X, X d T ) in T from Theorem A(iii).By the definition of ask zeta functions in §1.3, (1.2) is satisfied for all odd prime powers q .It is a simple exercise to show that since f n,S ( q ) is an integer for infinitely many q , therational function f n,S ( X ) is in fact a polynomial, as claimed. (cid:7) In the same way, parts (i)–(ii) of Theorem A imply analogous results for spaces ofgeneral n × m and antisymmetric n × n matrices with supports constrained by sets.Proposition 1.1 and Theorem A(ii) imply the following group-theoretic result (see §3.4). Corollary B (Class counting zeta functions of graphical group schemes) . Let Γ be a simple graph with m edges. Then, for each compact DVR O (of arbitrarycharacteristic) and with residue field size q , ζ cc G Γ ⊗ O ( s ) = W − Γ ( q, q m − s ) . In particular, graphical group schemes have strongly uniform class counting zeta functions.
As a very special case, we obtain the following consequence in the spirit of Higman’squestion on the class numbers k(U n ( F q )) for graphical groups over F q . Corollary 1.3.
Let Γ be a simple graph. Then there exists a polynomial f Γ ( X ) ∈ Q [ X ] such that, for all prime powers q , we have k( G Γ ( F q )) = f Γ ( q ) .Proof. We may take f Γ ( X ) ∈ Q ( X ) to be the coefficient of T of the rational power series W − Γ ( X, X m T ) in T . As in the proof of Corollary 1.2, f Γ ( X ) is a polynomial. (cid:7) Ingredients of the proof of Theorem A.
While our proof of Theorem A(i) can berecast in terms of existing machinery from the theory of zeta functions (“monomial p -adicintegrals” as in §1.6), parts (ii)–(iii) involve the development of several new tools thatare likely to have further applications beyond the present article. These include (a) anew type of zeta function associated with modules over polynomial rings (see §2) and,more generally, over toric rings (see §4), (b) a notion of “torically combinatorial” modules(see §4.4) which provides an algebraic explanation of uniformity, and (c) a novel blend ofgraph theory and toric geometry in §6.We note that the first author previously used toric geometry in the study of zetafunctions of groups and related structures; see [54, 55]. However, in that work, it turnedout to be extremely challenging to characterise those groups or algebras that are amenableto toric methods. In contrast, in the present setting, every graph provides an example ofsuch a group (scheme) via Theorem A(ii) and Corollary B.9 Introduction
Beyond uniformity.
Apart from being surprising in light of what is known about rankdistributions and matrices with restricted support, Theorem A also raises intriguingfollow-up questions. Which general features do the rational functions W ± Γ ( X, T ) and W H ( X, T ) possess? How do they depend on the graph Γ and hypergraph H , respectively?Do they afford a meaningful combinatorial interpretation? Can they be computed?Our proof of Theorem A is constructive and will thus provide an affirmative answerto the last of these questions. Regarding the first question, general results on ask zetafunctions from [57] have consequences such as the following: Corollary 1.4 (Functional equations) . Let W ( X, T ) be one of the rational functions W H ( X, T ) or W ± Γ ( X, T ) associated with a hypergraph or graph on n vertices. Then: W ( X − , T − ) = − X n T W ( X, T ) . Proof.
Combine [57, Theorem 4.18] and [59, §4]. (cid:7)
Corollary 1.5 (Reduced zeta functions) . Let the notation be as in Corollary 1.4. Then W (1 , T ) = 1 / (1 − T ) .Proof. Apply [57, §4.6]. (cid:7)
Theorem A and general results on zeta functions of algebraic structures (cf. e.g. [57,Theorem 4.10]) imply that each of the rational functions in Theorem A can be writtenin the form f ( X, T ) /g ( X, T ), where f ( X, T ) ∈ Q [ X ± , T ] and g ( X, T ) is a product offactors of the form 1 − X a T b for a, b ∈ Z with b (cid:62)
0. As we will see, we can oftenbe much more precise here. In particular, our next main results will cast light on therational functions W H ( X, T ) for arbitrary hypergraphs and on the rational functions W − Γ ( X, T ) (and hence associated class counting zeta functions) for certain graphs, namelythe so-called cographs . While constructive, the intricate recursive nature of our proof of Theorem A(ii)–(iii)provides few indications as to how the rational functions obtained depend on the graphin question. In contrast, in the case of hypergraphs we make the uniformity statement inTheorem A(i) fully explicit, as our next main result shows.Up to isomorphism, a hypergraph H as in §1.4 is completely determined by a vertexset V and, for each subset I ⊂ V , a “hyperedge multiplicity” µ I which counts how manyhyperedges of H have support I . We can explicitly describe W H ( X, T ) in terms of thesemultiplicities. Let d WO( V ) denote the poset of flags of subsets of V , i.e. (essentially) theposet of weak orders on V ; see Definition 5.3. Theorem C (Ask zeta functions of hypergraphs and weak orders) . Let H be a hypergraphwith vertex set V and given by a family µ = ( µ I ) I ⊂ V ∈ N P ( V )0 of hyperedge multiplicities.Then W H ( X, T ) = X y ∈ d WO( V ) (1 − X − ) | sup( y ) | Y J ∈ y X | J |− P I ∩ J = ∅ µ I T − X | J |− P I ∩ J = ∅ µ I T . (1.3)10
Introduction
The number of summands in (1.3) grows rather quickly. Indeed, let n = | V | . Asexplained in Remark 5.4, | d WO( V ) | = 4 f n , where f n is the n th Fubini (or ordered Bell)number, enumerating weak orders on V . In particular, f n ∼ n !2(log 2) n +1 (1.4)grows super-exponentially as a function of n ; see [3] or [72, §5.2]. The value of Theorem Clies not primarily in providing an algorithm for computing W H ( X, T ) but in the richcombinatorial structure of these functions that it reveals.We note that the right-hand side of (1.3) is similar but not identical to the “weakorder zeta functions” of Carnevale et al. [17, §1.2].
Consequences.
We exhibit three main applications of Theorem C. First, it imposessevere restrictions on the denominators of the functions W H ( X, T ). This turns out tohave remarkable consequences for analytic properties of ask zeta functions associatedwith hypergraphs; see Theorem 5.26. Secondly, for specific families of hypergraphs ofspecial interest here, we will obtain more manageable versions of Theorem C; see §§5.1.1,5.2.1, and 5.3.1. Finally, Theorem C will allow us to capture the effects of several naturaloperations for hypergraphs on the level of the rational functions W H ( X, T ); see §5.4.This will prove to be particularly valuable when combined with the results in §1.7.
Ingredients of the proof of Theorem C.
Let O be a compact DVR . Beginning withthe integral formalism for ask zeta functions from [57], our proof of Theorem C is basedon a formula of the same type as (1.3) for multivariate monomial integrals such as Z ( s ) := Z O n × O Y I = { i ,i ,... }⊂{ ,...,n } k x i , x i , . . . ; y k s I d µ ( x, y ) , (1.5)where s = ( s I ) I ⊂{ ,...,n } is a family of complex variables, k · k denotes the (suitablynormalised) maximum norm, and µ denotes the additive Haar measure on O n +1 with µ ( O n +1 ) = 1; see Theorem 5.5.Weak orders on a set encode the possible rankings of its elements that allow for ties.Given non-zero x , . . . , x n , y ∈ O , their valuations give rise to such a ranking via theusual order. In this way, weak orders naturally arise in the study of the integrals (1.5). Most of what we will learn about ask zeta functions associated with hypergraphs restsupon explicit formula such as (1.3). As indicated above, the starting point of theseformulae is an expression for the local ask zeta functions (i.e. those over compact
DVR s)associated with a hypergraph by means of a monomial integral as in (1.5). We have noreason to expect that such an approach will succeed for adjacency representations ofgraphs. (Example 7.5 will show that the integrals in (1.5) cannot suffice.) This explainswhy our proof of parts (ii)–(iii) of Theorem A is vastly more involved than that of part (i).11
Introduction
Our next main result exhibits a miraculous connection between the rational functions W − Γ ( X, T ) associated with certain simple graphs and the rational functions W H ( X, T )associated with hypergraphs in Theorem C.
Cographs.
The class of graphs known as cographs admits numerous equivalent char-acterisations; see §7.1. For instance, it is the smallest class of graphs which containsan isolated vertex and which is closed under both disjoint unions (denoted by ⊕ ) and“joins” (denoted by ∨ ) of graphs; here, the join of two graphs Γ and Γ is obtained fromtheir disjoint union by inserting edges connecting each vertex of Γ to each vertex of Γ .Equivalently, cographs are precisely those graphs that do not contain a path on fourvertices as an induced subgraph. Theorem D (Cograph Modelling Theorem) . Let Γ be a cograph. Then there exists anexplicit hypergraph H on the same vertex set as Γ such that W − Γ ( X, T ) = W H ( X, T ) . Informally, we think of the hypergraph H in Theorem D as a “model” of Γ in the sensethat, through the techniques that we developed here, the former allows us to determineand study the rational function W − Γ ( X, T ) much more easily than by the using methodsunderpinning Theorem A(ii). In particular, for a cograph Γ, Theorem D allows to express W − Γ ( X, T ) via Theorem C. We will construct a particular hypergraph H as in Theorem Dfor each cograph Γ; we refer to this hypergraph as “the” model of Γ in the following.Our construction reveals a number of specific properties of models. For instance,models always have fewer hyperedges than vertices. Moreover, the sum over the entriesof an incidence matrix of a model is always even (this will follow from Remark 7.25), justas for graphs. These conditions further illustrate the level of generality of Theorem C.We note that the special case of Theorem D obtained by taking Γ to be a completegraph Γ on n vertices and H to be a hypergraph on n vertices with n − Ingredients of the proof of Theorem D.
In the same way that our proof of Theorem Agoes beyond merely establishing uniformity of zeta functions by elucidating the structureof certain modules, the cograph modelling theorem is based on more than a merecoincidence of rational functions. Instead, it is a consequence of a structural counterpart(Theorem 7.1) of Theorem D which establishes that for each cograph Γ, there existsan (explicit) hypergraph H such that the “negative adjacency module” of Γ and the“incidence module” of H , while generally non-isomorphic, are “torically isomorphic” (upto a well-understood direct summand). Our proof of this fact involves once again a blendof graph theory and toric geometry. We note that we have found no evidence that wouldpoint towards a modelling theorem for the rational functions W +Γ ( X, T ) associated withan interesting class of (not necessarily simple) graphs Γ.12
Introduction
Group-theoretic applications.
By a cographical group (scheme) , we mean a graph-ical group (scheme) (see §1.4) arising from a cograph. By combining Corollary B,Theorem C, and Theorem D, we obtain an explicit formula for (local) class counting zetafunctions of cographical group schemes in terms of the associated modelling hypergraphs.In particular, many of our results on ask zeta functions of hypergraphs (e.g. explicitformulae and information on analytic properties) have immediate applications to the classcounting zeta functions of the associated cographical group schemes. These are recordedin §8. For instance, as a substantial generalisation of several previously known formulae,we explicitly determine the (local) class counting zeta functions of the cographical groupschemes associated with the following classes of cographical groups over Z :(i) The class of finite direct products of finitely generated free class-2-nilpotent groups.(ii) The class of class-2-nilpotent free products of free abelian groups of finite rank.(iii) The smallest class of groups which contains Z and which is closed under both directproducts with Z and class-2-nilpotent free products with Z .The class counting zeta functions of the cographical group schemes associated with freeclass-2-nilpotent groups and class-2-nilpotent free products of two free abelian groupshave been previously determined by Lins [47, Corollary 1.5].As we noted above, right-angled Artin groups are close relatives of our graphical groups.Right-angled Artin groups associated with cographs have e.g. been studied in [38, 63]. We illustrate Theorems A, C, and D by means of a simple yet instructive example thatwe will repeatedly revisit throughout this paper.
Example 1.6.
Let Γ be the following simple graph:Using the constructive arguments underpinning Theorem A(ii)–(iii), we may explicitlycompute the rational functions W ± Γ ( X, T ) (see §9): W − Γ ( X, T ) = 1 + X − T − X − T − X − T + X − T + X − T (1 − T ) (1 − XT ) and (1.6) W +Γ ( X, T ) = F ( X, T ) / ((1 − X − T )(1 − X − T )(1 − X − T )(1 − X − T )(1 − X − T )(1 − X − T ) (1 − T − X )(1 − T ) ) , (1.7)13 Introduction where the (unwieldy) numerator F ( X, T ) of W +Γ ( X, T ) is recorded in Table 5 on p. 103.Alternatively, the first of these rational functions can be found using Theorems C–D.Indeed, Γ is a cograph for the subgraph induced by all vertices excluding those twodepicted on the central horizontal edge is a disjoint union of two complete graphs onthree vertices each. As the aforementioned central vertices are connected to all othervertices, it follows that Γ is a cograph; in fact, we have just shown that Γ is isomorphicto (K ⊕ K ) ∨ K , where K n denotes the complete graph with vertices 1 , . . . , n .Let H be a hypergraph on 8 vertices with 7 hyperedges and incidence matrix ∈ M × ( Z ) . (1.8)Write [ n ] = { , . . . , n } . Using the notion of hyperedge multiplicities from §1.6, up toisomorphism, H is thus given by the family µ = ( µ I ) I ⊂ [8] with µ [8] = µ [5] = µ { , , , , } = 2 , µ [2] = 1 , and µ I = 0 for all remaining subsets I ⊂ [8]. Then the explicit form of Theorem D (see §7)shows that W − Γ ( X, T ) = W H ( X, T ); see Example 7.28. In particular, the formula (1.6)for W − Γ ( X, T ) is, in principle, given by Theorem C, as a sum indexed by the poset d WO([8]). Rather than handle a sum over the 2 , ,
340 elements of this poset directly,it is far more convenient to apply some of the tools for recursively computing ask zetafunctions associated with hypergraphs that we will develop in §5. For details of this shortcomputation of W H ( X, T ), see Example 5.25.
We collect consequences of our main results from above—to be proved in §8.1—that areboth closely related to topics of interest in asymptotic and finite group theory and thatseem likely to provide promising avenues for fruitful further research.
Non-negativity.
Let Γ be a simple graph with m edges. Let W − Γ ( X, T ) be as inTheorem A(ii) and expand W − Γ ( X, X m T ) = ∞ X k =0 f Γ ,k ( X ) T k for f Γ ,k ( X ) ∈ Q ( X ). By Corollary B, f Γ ,k ( q ) is the class number of the graphical group G Γ ( O / P k ) for each compact DVR O with maximal ideal P and residue field size q .14 Introduction
In particular, f Γ , ( X ) is precisely the polynomial (!) that we denoted by f Γ ( X ) inCorollary 1.3. Our proof of the latter result implies that, in fact, each f Γ ,k ( X ) is apolynomial in X . Inspired by Lehrer’s conjecture [43] on character degrees and similarresults on class numbers of the groups U n ( F q ) by Vera-López et al. (see, for instance, [69]),both refinements of Higman’s conjecture from §1.2, we obtain the following. Theorem E.
Let Γ be a cograph. Then the coefficients of each f Γ ,k ( X ) as a polynomialin X − are non-negative. Question 1.7.
For which simple graphs Γ does the conclusion of Theorem E hold?
Analytic properties.
The most fundamental analytic invariant of a (non-negative)Dirichlet series is its abscissa of convergence which encodes the precise degree of polynomialgrowth of the series’s partial sums. In a seminal paper [24], du Sautoy and Grunewaldshowed that subgroup zeta functions associated with nilpotent groups have rationalabscissae of convergence. The same turns out to be true for class counting zeta functionsof arbitrary Baer group schemes. (For a proof, combine Proposition 2.5 below and[57, Theorem 4.20].) For cographical group schemes, we can do much better.
Theorem F.
Let Γ be a cograph with n vertices and m edges. There exists a positiveinteger α (Γ) (cid:54) n + m + 1 such that if O K is the ring of integers of an arbitrary numberfield K , then the abscissa of convergence of ζ cc G Γ ⊗O K ( s ) is equal to α (Γ) . Moreover, if O is a compact DVR , then the real part of each pole of ζ cc G Γ ⊗ O ( s ) is a positive integer. By [57, Theorem 4.20], for an arbitrary simple graph Γ, there is a (unique) positiverational number α (Γ) with properties as in Theorem F. However, it is not clear if α (Γ) isalways an integer. The positivity of local poles in Theorem F is related to [57, Question 9.4].The integrality of local poles in Theorem F does not carry over to arbitrary graphs. Forinstance, for the graph Γ in Example 7.5, the function ζ cc G Γ ⊗ O ( s ) has a pole at 3 / Question 1.8.
Let Γ be a simple graph.(i) Is α (Γ) always an integer?(ii) Are the real parts of the poles of ζ cc G Γ ⊗ O ( s ) for compact DVR s O always half-integers?(iii) Is there are a meaningful combinatorial formula (in the spirit of Theorem C) forthe functions W ± Γ ( X, T ) which is valid for all graphs on a given vertex set?(iv) What do the numbers α (Γ) and the poles of class counting zeta functions ofgraphical group schemes tell us about a graph? How are they related to othergraph-theoretic invariants? Section 2.
In §2, we collect basic facts about ask zeta functions including, in particular,the crucial duality operations from [60]. Along the way, in §2.4, we formally define15
Introduction
Baer group schemes and relate their class counting zeta functions to ask zeta functionsattached to alternating bilinear maps. Apart from reviewing background material, wealso develop a “cokernel formalism” (see §2.5) for expressing ask zeta functions in termsof p -adic integrals. In §2.6, we use this to interpret ask zeta functions as special cases ofa more general class of zeta functions attached to modules over polynomial rings. Section 3.
After reviewing basic constructions and terminology pertaining to graphs andhypergraphs in §3.1 we define, in §§3.2–3.3, the adjacency and incidence representationsinformally described in §1.4. We further define adjacency and incidence modules andrelate their zeta functions in the sense of §2.6 to the ask zeta functions associated withadjacency and incidence representations. In §3.4, we formally define graphical groups andgroup schemes and relate class counting zeta functions of the latter to ask zeta functionsof adjacency representations.
Section 4.
Toric geometry enters the scene in §4. We begin by collecting basic factsfrom convex geometry in §4.1 and on toric rings and schemes in §4.2. In §4.3, we furtherenlarge the class of zeta functions introduced in §2.6 (which, as we saw, includes askzeta functions) by attaching zeta functions to modules over toric rings. In §4.4, we proveTheorem A(i) and introduce the key concept of “torically combinatorial” modules thatwill also form the basis of our proof of Theorem A(i)–(ii).
Section 5. §5 is devoted to a detailed analysis of the rational functions W H ( X, T )attached to hypergraphs H via Theorem A(i). In §5.1, we prove (a slightly more generalversion of) Theorem C. The remainder of §5 then focuses on two main themes. First,for several classes of hypergraphs of interest, we provide more manageable forms ofTheorem A(i). These classes are the “staircase hypergraphs” in §5.1.1, disjoint unionsof “block hypergraphs” in §5.2.1, and the “reflections” of the latter family in §5.3.1.Secondly, as we will explore and exploit throughout §5.2–5.4, the general formula providedby Theorem C behaves very well with respect to natural operations on hypergraphs.Finally we deduce, in §5.5, consequences for analytic properties of ask zeta functions ofhypergraphs.Later on, our results from §5 will find group-theoretic applications in §8 via theCograph Modelling Theorem (Theorem D, proved in §7). In particular, the hypergraphoperations alluded to above will translate to natural group-theoretic operations. Section 6.
In §6, we prove Theorem A(ii)–(iii) and also Corollary B. Our proof considerspositive and negative adjacency representations of graphs simultaneously by means of acommon generalisation, the “weighted signed multigraphs” (
WSM s) introduced in §6.1.Multigraphs are more general than graphs in that they allow parallel edges. Each
WSM gives rise to an adjacency module (over a suitable toric ring) which generalises the positiveand negative adjacency modules of graphs from §3. In §6.2, we describe a number of“surgical procedures” for
WSM s that do not affect the associated adjacency modules.Even when the original multigraph was a graph, these procedures may introduce parallel16
Introduction edges—this justifies introducing the concept of
WSM s. After some technical preparationsin §6.3, we use these procedures in §6.4 to give an inductive proof of Theorem A(ii)–(iii).
Section 7.
The Cograph Modelling Theorem (Theorem D) is the subject of §7. Wefirst recall basic facts about cographs in §7.1. In §7.2, we then explain how Theorem Dfollows from a structural comparison result (Theorem 7.1) relating adjacency modules ofcographs and incidence modules of hypergraphs. Extending upon ideas underpinning theproof of Theorem A(ii)–(iii), the remainder of §7 is then devoted to proving Theorem 7.1.An overview of the ingredients featuring in our proof and of our overall strategy is givenin §7.3. This is followed by an implementation of this strategy in §§7.4–7.7.
Section 8.
In this section we combine Theorem A, Corollary B, Theorem C, Theorem D,and our further analysis of the rational functions W H ( X, T ) associated with hypergraphsfrom §5 to deduce structural properties and produce explicit formulae for class countingzeta functions associated with “cographical group schemes”, i.e. graphical group schemesarising from cographs. We consider, in particular, the cographical group schemes associ-ated with the families of nilpotent groups listed in the final part of §1.7 and also relateour results to work of Lins [46, 47] on bivariate conjugacy class zeta functions.
Section 9–10.
Based on our constructive proof of Theorem A and computationaltechniques developed by the first author, in §9, we provide further examples of therational functions W ± Γ ( X, T ) associated with graphs Γ on few vertices. Many of theseexamples are not covered by Theorems C–D. Motivated by such computational evidence,in §10, we pose and discus a number of questions for further research beyond thosealready mentioned in §1.9.
Sets.
The symbol “ ⊂ ” signifies not necessarily strict inclusion. We write t for thedisjoint union (= coproduct) of sets. Throughout, V denotes a finite set, typically ofvertices of a graph or hypergraph and of cardinality n . The power set of V is denotedby P ( V ). We write N = { , , . . . } and N = N ∪ { } . The complement of I within someambient set V is denoted by I c := V \ I . We write [ n ] = { , , . . . , n } and [ n ] = [ n ] ∪ { } . Rings and modules.
Rings are assumed to be associative, commutative, and unital.Let R be a ring. The unit group of R is denoted by R × . The dual of an R -module M is M ∗ = Hom( M, R ). An R -algebra consists of a ring S together with a ring map R → S .For a set V , let RV = L v ∈ V Rv be the free module on V ; we extend this notation tosubsets of R in the evident way and e.g. write R (cid:62) V = ( P v ∈ V λ v v : ∀ v ∈ V.λ v ∈ R (cid:62) ) ⊂ R V , where R (cid:62) = { x ∈ R : x (cid:62) } . For x ∈ RV , we use the suggestive notation x =17 Introduction P v ∈ V x v v = ( x v ) v ∈ V . Often, X = ( X v ) v ∈ V denotes a family of algebraically independentelements over R .We let M n × m ( R ) (resp. M n ( R )) denote the module of all n × m (resp. n × n ) matricesover R . The transpose of a matrix a is denoted by a > . Discrete valuation rings.
Throughout this article, O denotes a discrete valuation ring( DVR ) with maximal ideal P . We write O k = O / P k and ( · ) k = ( · ) ⊗ O k . Let q = | O / P | denote the size of the residue field of O . We write ( k ) for the number 1 − q − k .Let ν : O → N ∪ {∞} denote the (surjective) normalised valuation on O and let | · | be the absolute value | a | = q − ν ( a ) on O . For a non-empty collection C of elements of O ,write k C k = max {| a | : a ∈ C } . For a free O -module M of finite rank, µ M denotes theadditive Haar measure on M with µ ( M ) = 1.For a non-zero O -module M , we write M × = M \ P M ; we let { } × = { } . Miscellaneous.
Maps usually act on the right. In particular, we regard an n × m matrixover a ring R as a linear map R n → R m .The n × n identity matrix is denoted by 1 n . In contrast, n × m (and n = n × n ) denotesthe respective all-one matrix. The all-one vector of length n is denoted ( n ) . The freenilpotent group of class at most c on d generators is denoted by F c,d .We write δ ij for the usual “Kronecker delta”; more generally, for a Boolean value P ,we let δ P = 1 if P is true and δ P = 0 otherwise. Further notation
Notation comment reference ζ cc G ( s ) class counting zeta function §1.3 W H ( X, T ), W ± Γ ( X, T ) ask zeta functions of (hyper)graphs Thm A θ S , η O , γ R ± , . . . base change of module representations §2.1 θ ◦ , θ • Knuth duals §2.1 A U,V,Wθ ( Z ), C U,V,Wθ ( X ) matrices of a module representations §2.2ask( θ ), ζ ask θ ( s ), Z ask θ ( T ) average size of kernel, ask zeta functions §2.3 ζ M ( s ) zeta function associated with a module §§2.6, 4.3V( H ), E( H ) vertex and (hyper)edge set of a (hyper)graph §3.1 v ∼ v , v ∼ e adjacency and incidence relation §3.1 H ( µ ), H ( V | P µ I I ) hypergraph with given hyperedge multiplicities Def. 3.1 H ⊕ H , H (cid:16) H disjoint resp. complete union §3.1 inc , Inc, η incidence modules and representations §3.2 adj , Adj, γ ± adjacency modules and representations §§3.3, 7.5 G (cid:16) G free class-2-nilpotent product of groups (3.11) G Γ graphical group scheme §3.4 |F | support of a fan §4.1 F ∧ F coarsest common refinement of fans §4.1 (cid:54) σ preorder defined by a cone §4.118 Ask zeta functions and modules over polynomial rings R σ toric ring §4.2 σ ( O ) “rational points” of a cone over a DVR §4.1gp ( x ) (resp. gp ( x )) x − x (resp. − x ) §5.1 d WO( V ), g WO( V ) weak orders Def. 5.3 F ( T ) ? G ( T ) Hadamard product §5.2 H , H , H , H insert all-one or all-zero row or column Def. 5.22 Γ weighted signed multigraph ( WSM ) §6.1 H ( S ), Γ ( S ) hypergraph and WSM defined by a scaffold §7.5Kite( k ) kite graph §8.4 In this section, we recall background material on module representations and associatedask zeta functions from [57, 60]. We also relate the latter functions to class counting zetafunctions associated with Baer group schemes. Finally, we develop a “cokernel formalism”for ask zeta functions which allows us to view the latter as special cases of a more generalclass of functions attached to modules over polynomial rings.Throughout, let R be a ring. In §1.3, we attached ask zeta functions to module homomorphisms M → M n × m ( R ).Rather than focusing on such parameterisations of modules of matrices, we will use thefollowing coordinate-free approach from [60, §2]; as shown in [60], disposing of coordinateselucidates duality phenomena.By a module representation over R , we mean a homomorphism A θ −→ Hom(
B, C ),where A , B , and C are R -modules. Base change.
For a ring map R λ −→ S , an R -module M , and an S -module N , we let M λ = M ⊗ R S (resp. N λ ) denote the extension (resp. restriction) of scalars of M (resp. N )along λ ; this is an S -module (resp. an R -module). When the reference to λ is clear, wesimply write M S = M λ and N R = N λ .Let A θ −→ Hom(
B, C ) be a module representation over R . Given a ring map R λ −→ S ,extension of scalars along λ yields a module representation A λ θ λ −→ Hom( B λ , C λ )over S . When the reference to λ is clear, we write θ S = θ λ . Knuth duals.
Given a module representation A θ −→ Hom(
B, C ), let θ ◦ denote the modulerepresentation B → Hom(
A, C ) with a ( bθ ◦ ) = b ( aθ ) for a ∈ A and b ∈ B .19 Ask zeta functions and modules over polynomial rings
Write ( · ) ∗ = Hom( · , R ). Apart from θ ◦ , the module representation θ also gives rise toa module representation C ∗ θ • −→ Hom(
B, A ∗ ) defined by a ( b ( ψθ • )) = ( b ( aθ )) ψ for a ∈ A , b ∈ B , and ψ ∈ C ∗ .Note that ( θ λ ) ◦ = ( θ ◦ ) λ for each ring map R λ −→ S . Moreover, if A and C are bothfinitely generated and projective, then we may identify ( θ • ) λ = ( θ λ ) • . For more on theoperations θ θ ◦ and θ θ • , see [60, §§4–5]. Direct sums, homotopy, and isotopy.
Let A θ −→ Hom(
B, C ) and A θ −→ Hom( B , C ) bemodule representations over R . The direct sum of θ and θ is the module representation A ⊕ A θ ⊕ θ −−−→ Hom( B ⊕ B , C ⊕ C ) , ( a, a ) aθ ⊕ a θ . A homotopy θ → θ is a triple of homomorphisms ( A α −→ A , B β −→ B , C γ −→ C ) suchthat the following diagram commutes for each a ∈ A : B aθ −−−−→ C β y y γ B ( aα ) θ −−−−→ C . Module representations over R together with homotopies as morphisms naturally form acategory. An invertible homotopy is called an isotopy . Up to isotopy, a module representation involving free modules of finite rank can beequivalently expressed in terms of a matrix of linear forms. In detail, let A θ −→ Hom(
B, C )be a module representation over R . Suppose that each A , B , and C is free of finite rank.By choosing bases U , V , and W of A , B , and C , respectively, we may identify A = RU , B = RV , and C = RW .Let Z = ( Z u ) u ∈ U consist of algebraically independent variables over R . Define an R [ Z ]-linear map A U,V,Wθ ( Z ) := (cid:16) X u ∈ U Z u u (cid:17) θ R [ Z ] ∈ Hom( R [ Z ] V, R [ Z ] W ) . Informally, A U,V,Wθ ( Z ) is the image of a “generic element” of A = RU under θ . The matrixof A U,V,Wθ ( Z ) with respect to the bases V and W of R [ Z ] V and R [ Z ] W , respectively, isthe matrix of linear forms associated with θ (and the chosen bases) from [60, §4.4].As we will now explain, by specialising A U,V,Wθ ( Z ), we may recover θ (and θ λ for eachring map R λ −→ S ). Lemma 2.1.
Let S be an R -algebra and let z ∈ SU . Let S z denote S regarded as an R [ Z ] -algebra via Z u s = z u s ( u ∈ U, s ∈ S ). Ask zeta functions and modules over polynomial rings (i) A U,V,Wθ ( z ) := A U,V,Wθ ( Z ) ⊗ R [ Z ] S z ∈ Hom(
SV, SW ) coincides with zθ S .(ii) Coker( zθ S ) ≈ S Coker( A U,V,Wθ ( Z )) ⊗ R [ Z ] S z .Proof. Part (i) is clear. Part (ii) follows from (i) and right exactness of tensor products. (cid:7)
Remark 2.2.
The isomorphism types of the cokernels in Lemma 2.1(ii) (over S and R [ Z ], respectively) clearly only depend on the isotopy type (see §2.1) of θ .Recall the definition of θ ◦ from §2.1. In subsequent sections, we will analyse andcompute certain zeta functions associated with θ by studying the sizes of Coker( x ( θ ◦ ) S )for suitable finite R -algebras S and x ∈ SV . For this purpose, it will be convenient to useexplicit (finite) presentations of these cokernels. Let X = ( X v ) v ∈ V consist of algebraicallyindependent variables over R . In accordance with the notation in [57, §4.3.5], let C U,V,Wθ ( X ) := A V,U,Wθ ◦ ( X ) = X v ∈ V X v v ! ( θ ◦ ) R [ X ] ∈ Hom ( R [ X ] U, R [ X ] W ) . (2.1)Informally, the image of C U,V,Wθ ( X ) is the “additive orbit” xM = { xa : a ∈ M } , where x is a “generic element” of RV and M denotes the image of θ . We can read off explicitgenerators for the image of C U,V,Wθ ( X ) and hence a presentation for the latter’s cokernel. Lemma 2.3.
Im( C U,V,Wθ ( X )) = *(cid:16) X v ∈ V X v v (cid:17) ( uθ R [ X ] ) : u ∈ U + . (cid:7) Note that we may identify uθ R [ X ] = uθ ⊗ R R [ X ] for u ∈ U . Let A θ −→ Hom(
B, C ) be a module representation over a ring R . If A and B are bothfinite as sets, then the average size of the kernel of A acting on B via θ is the numberask( θ ) := 1 | A | X a ∈ A | Ker( aθ ) | . Suppose that R admits only finitely many ideals of a given finite index. Furthersuppose that A , B , and C are finitely generated. The (analytic) ask zeta function associated with θ is the Dirichlet series ζ ask θ ( s ) := X I/R ask( θ R/I ) | R/I | − s , where s is a complex variable and the summation extends over those ideals I / R with | R/I | < ∞ . If R is the ring of integers of a global or local field, then ζ ask θ ( s ) defines ananalytic function on a right half-plane { s ∈ C : Re( s ) > α } ; cf. [57, §§3.2–3.3]. Theinfimum of all such real numbers α > abscissa of convergence α θ of ζ ask θ ( s ).21 Ask zeta functions and modules over polynomial rings
Let O be a compact DVR with maximal ideal P . We write O k = O / P k and, moregenerally, ( · ) k = ( · ) ⊗ O O k . Let A θ −→ Hom(
B, C ) be a module representation over O .Suppose that each of A , B , and C is free of finite rank. We may identify θ k = θ ⊗ O O k and θ O k (see §2.1). Consider the generating function Z ask θ ( T ) := ∞ X k =0 ask( θ k ) T k . By slight abuse of terminology, we also refer to Z ask θ ( T ) as the (algebraic) ask zetafunction of θ . By [57, Theorem 1.4], if O has characteristic zero, then Z ask θ ( T ) ∈ Q ( T ).Note that the analytic function ζ ask θ ( s ) and its algebraic counterpart Z ask θ ( T ) determineone another: ζ ask θ ( s ) = Z ask θ ( q − s ), where q denotes the residue field size of O . As explainedin [57, §8], ask zeta functions (of either type) arise naturally in the enumeration of orbitsand conjugacy classes of unipotent groups. Most of the main results of this article(e.g. Theorems A,C–D) are stated in terms of the generating functions Z ask θ ( T ) while theanalytic functions ζ ask θ ( s ) feature in our proofs by means of suitable p -adic integrals.We will rely heavily on the fact that the duality operations θ θ ◦ and θ θ • (see §2.1)have the following tame effects on ask zeta functions. Theorem 2.4 ([60, Corollary 5.6]) . Z ask θ ( T ) = Z ask θ ◦ (cid:16) q rk( B ) − rk( A ) T (cid:17) = Z ask θ • ( T ) . Consider an alternating bilinear map (cid:5) : A × A → B of Z -modules. Suppose that B isuniquely 2-divisible (i.e. a Z [1 / Baer group associated with (cid:5) is thenilpotent group G (cid:5) of class at most 2 on the set A × B with multiplication( a, b ) ∗ ( a , b ) = (cid:16) a + a , b + b + 12 ( a (cid:5) a ) (cid:17) ;this construction is part of the Baer correspondence [2]. We now describe a versionof the operation (cid:5) (cid:32) G (cid:5) for group schemes.Let V and E be disjoint finite sets and let (cid:5) : Z V × Z V → Z E be an alternatingbilinear map. We obtain a nilpotent Lie Z -algebra (“Lie ring”) g (cid:5) of class at most 2 withunderlying Z -module Z V ⊕ Z E and Lie bracket [ x + c, y + d ] = x (cid:5) y for x, y ∈ Z V and c, d ∈ Z E .Let v be a total order on V ∪ E . By [66, §2.4.1], there exists a (unipotent) groupscheme G (cid:5) = G (cid:5) , v over Z such that for each ring R , we may identify G (cid:5) ( R ) = RV ⊕ RE as sets and such that group commutators in G (cid:5) ( R ) coincide with Lie brackets in g (cid:5) ⊗ Z R .The multiplication ∗ on G (cid:5) ( R ) is characterised by the following properties:(i) For v v · · · v v k in V and r , . . . , r k ∈ R , ( r v ) ∗ · · · ∗ ( r k v k ) = r v + · · · + r k v k .(ii) For v, w ∈ V with v v w and r, s ∈ R , we have ( sw ) ∗ ( rv ) = rv + sw − rs ( v (cid:5) w ).(iii) RE is a central subgroup of G (cid:5) ( R ) and x ∗ c = x + c for x ∈ G (cid:5) ( R ) and c ∈ RE .22 Ask zeta functions and modules over polynomial rings
Up to isomorphism, G (cid:5) does not depend on v . We call G (cid:5) the Baer group scheme associated with (cid:5) . For each ring R in which 2 is invertible, G (cid:5) ( R ) is isomorphic to theBaer group attached to the alternating bilinear map RV × RV → RE obtained from (cid:5) . Proposition 2.5.
Let (cid:5) : Z V × Z V → Z E be an alternating bilinear map. Let Z V α −→ Hom( Z V, Z E ) be the module representation with v ( wα ) = v (cid:5) w for v, w ∈ V . Let m = | E | .Let R be the ring of integers of a local or global field of arbitrary characteristic. Let G (cid:5) be the Baer group scheme associated with (cid:5) as above. Then ζ cc G (cid:5) ⊗ R ( s ) = ζ ask α R ( s − m ) .Proof. For each finite ring A , commutators in G (cid:5) ( A ) are given by Lie brackets in g (cid:5) ⊗ Z A .By the same reasoning as in [60, Lemma 7.1], we see that k( G (cid:5) ( A )) = | A | m ask( α A ). (cid:7) Proof of Proposition 1.1.
Let (cid:5) : Z n × Z n → M ∗ be the alternating bilinear map givenby a ( x (cid:5) y ) = xay > from §1.2. Let Z n α −→ Hom( Z n , M ∗ ) with x ( yα ) = x (cid:5) y . ByProposition 2.5, ζ cc G M ⊗ R ( s ) = ζ ask α R ( s − m ). We claim that α • (see §2.1) is isotopic to M ι , −→ so n ( Z ). By [60, Proposition 4.8], this is equivalent to ι • being isotopic to α whichis easily verified using the isomorphism ( Z n ) ∗ ≈ Z n given by matrix transposition. UsingTheorem 2.4, we conclude that ζ ask α R ( s ) = ζ ask ι R ( s ) provided that the field of fractions of R is a local field. Finally, using [60, Remark 5.5], the global case reduces to the localone. (cid:7) From now on, let O be a compact DVR . Let A θ −→ Hom(
B, C ) be a module representationover O , where each of A , B , and C is free of finite rank. For a ∈ A and y ∈ O , the map B aθ −→ C gives rise to an induced map B ⊗ O O /y aθ ⊗ O /y = ( a ⊗ θ O /y −−−−−−−−−−−−−→ C ⊗ O O /y . Define C θ ( a, y ) := | Coker( aθ ⊗ O /y ) | . The following is equivalent to [57, Theorem 4.5], but with kernels replaced by cokernels.
Theorem 2.6.
For
Re( s ) (cid:29) , ζ ask θ ( s ) = (1 − q − ) − Z A × O | y | s − rk( B )+rk( C ) − C θ ( a, y ) d µ A × O ( a, y ) . Proof.
As in [57, Definition 4.4] (see [60, §3.5]), write K θ ( a, y ) = | Ker( aθ ⊗ O /y ) | . Theclaim then follows immediately from [57, Theorem 4.5] (cf. [60, Theorem 3.5]) and the factthat by the first isomorphism theorem, K θ ( a, y ) = C θ ( a, y ) · | y | rk( C ) − rk( B ) for y = 0. (cid:7) Remark 2.7.
In the present article, we express ask zeta functions in terms of sizesof cokernels, rather than kernels; see Corollary 2.8. Cokernels turn out to be moreconvenient here since they commute with base change (both being colimits).23
Ask zeta functions and modules over polynomial rings
Explicit dual form of Theorem 2.6.
We can make Theorem 2.6 more explicit by choosingbases. Let A = O U , B = O V , and C = O W , where U , V , and W are finite sets. As in§2.2, let Z = ( Z u ) u ∈ U consist of algebraically independent elements. By Lemma 2.1, C θ ( a, y ) = | Coker( A U,V,Wθ ( Z ) ⊗ O [ Z ] ( O /y ) a ) | , (2.2)where ( O /y ) a denotes O /y with the O [ Z ]-module structure Z u r = a u r ( u ∈ U , r ∈ O /y ).It will be convenient to express ζ ask θ ( s ) in terms of θ ◦ via Theorem 2.4. Let X = ( X v ) v ∈ V consist of algebraically independent variables over O . Recall from (2.1) the definition ofthe O [ X ]-homomorphism C U,V,Wθ ( X ). Corollary 2.8.
For
Re( s ) (cid:29) , ζ ask θ ( s ) = (1 − q − ) − Z O V × O | y | s −| V | + | W |− | Coker( C U,V,Wθ ( X )) ⊗ O [ X ] ( O /y ) x | d µ O V × O ( x, y ) . Proof.
Combine Theorem 2.4, equation (2.2), and Theorem 2.6. (cid:7)
We will study ask zeta functions by considering a sequence of successive generalisationsof the integrals featuring in Corollary 2.8. As our first step in this direction, we considerintegrals obtained from Corollary 2.8 by allowing Coker( C U,V,Wθ ( X )) to be a more generaltype of O [ X ]-module.As before, we write X = ( X v ) v ∈ V , where the X v are algebraically independent variablesover O (or whichever base ring we consider).Let M be a finitely generated O [ X ]-module. The example of primary interest to us atthis point is the case M = Coker( C U,V,Wθ ( X )) (see (2.1)), where θ is a suitable modulerepresentation over O . As in §2.2, for x ∈ O V , let O x denote O endowed with the O [ X ]-module structure X v r = x v r ( v ∈ V , r ∈ O ). More generally, for an arbitrary O -module N , we let N x denote the O [ X ]-module N ⊗ O O x (cf. Lemma 2.10). Definition 2.9.
Define a zeta function ζ M ( s ) := Z O V × O | y | s − · | M x ⊗ O O /y | d µ O V × O ( x, y ) . (2.3)The following simple observation will be used repeatedly throughout this article. Lemma 2.10.
Let R be a ring. Let S and S be R -algebras. Let X = Spec( S ) /R andlet x ∈ X ( S ) . Let S χ −→ S be the R -algebra map corresponding to χ . Let M be an S -module and let M be an S -module. Let M x := ( M ) χ and M x := M χ (see §2.1);these are both ( S, S ) -bimodules. Then M x ⊗ S M and M ⊗ S M x both carry naturallyisomorphic ( S, S ) -bimodule structures.Proof. Apply [9, Ch. II, §3, no. 8] with A = S , B = S , E = M , F = S χ , and G = M . (cid:7) Modules and module representations from (hyper)graphs
Remark 2.11.
Note that for x ∈ O V = Spec( O [ X ])( O ), the definitions of the O -modules M x and ( O /y ) x provided Lemma 2.10 coincide with those given above. Inparticular, Lemma 2.10 allows us to identify M x ⊗ O O /y = M ⊗ O [ X ] ( O /y ) x in (2.3). (Fora proof, relabel M ↔ M and take S = O , S = O [ X ], and M = O /y in Lemma 2.10.) Remark 2.12. If O has characteristic zero, then standard arguments from p -adicintegration show that ζ M ( s ) ∈ Q ( q − s ) and also establish “Denef-type formulae” in aglobal setting; cf. [57, §4.3.3].The following is immediate from Corollary 2.8. Corollary 2.13.
Let U , V , and W be finite sets and X = ( X v ) v ∈ V as before. Let O U θ −→ Hom( O V, O W ) be a module representation over O . Let M = Coker( C U,V,Wθ ( X )) ;see (2.1) . Then ζ ask θ ( s ) = (1 − q − ) − ζ M ( s − | V | + | W | ) . (cid:7) In particular, the zeta functions attached to modules over polynomial rings in Defini-tion 2.9 generalise local ask zeta functions. We will further generalise the former functionsby suitably replacing polynomial rings by toric rings; see Definition 4.4. As we will see,this greater generality will provide us with the means to study “toric properties” of askzeta functions by purely combinatorial means.
In this section, we begin by fixing our notation for various concepts related to graphs andhypergraphs. Formalising and generalising our discussion from §1.4, for each hypergraph H and (simple) graph Γ, we define the incidence representation η of H and the adjacencyrepresentations γ ± of Γ, as well as corresponding incidence and adjacency modules Inc( H )and Adj(Γ; ± R be a ring. For a general reference on hypergraph theory, see e.g. [13].
Hypergraphs. A hypergraph is a triple H = ( V, E, | · | ) consisting of a finite set V of vertices , a finite set E of hyperedges , and a support function E | · | −→ P ( V ). Weoften tacitly assume that V ∩ E = ∅ . When confusion is unlikely, we often omit | · | (andoccasionally even V and E ) from our notation. We write V( H ) = V , E( H ) = E , and | · | H = | · | . Two hyperedges e and e are parallel if | e | = | e | . An edge is a hyperedge e with | e | ∈ { , } . An edge e with | e | = 1 is a loop . The reflection of H is thehypergraph H c = ( V, E, | · | c ) with | e | c = V \ | e | . An isomorphism between hypergraphs H and H consists of bijections V( H ) φ −→ V( H ) and E( H ) ψ −→ E( H ) such that | e | H P ( φ ) = | eψ | H for each e ∈ E( H ), where P ( φ ) is the direct image map P (V( H )) → P (V( H ))induced by φ . 25 Modules and module representations from (hyper)graphs
Incidence matrices.
Let H = ( V, E, | · | ) be a hypergraph. A vertex v ∈ V and hyperedge e ∈ E are incident , written v ∼ H e or simply v ∼ e , if v ∈ | e | . Let I ( H ) denote the setof all pairs ( v, e ) ∈ V × E with v ∼ H e . Write V = { v , . . . , v n } and E = { e , . . . , e m } ,where n = | V | and m = | E | . The incidence matrix of H with respect to the givenorderings of the elements of V and E is the n × m (0 , a ij ] with a ij = 1 if andonly if v i ∼ H e i . Hypergraph operations.
The disjoint union H ⊕ H of hypergraphs H i = ( V i , E i , | · | i )( i = 1 ,
2) is the hypergraph on the vertex set V t V (disjoint union) with hyperedge set E t E and support function k e i k = | e i | i for e i ∈ E i . If A i ∈ M n i × m i ( Z ) is an incidencematrix of H i , then the block-diagonal matrix " A A ∈ M ( n + n ) × ( m + m ) ( Z )is an incidence matrix of H ⊕ H .The complete union H (cid:16) H is the hypergraph on the vertex set V t V withhyperedge set E t E and support function k e i k = | e i | i t V j for e i ∈ E i and i + j = 3. If A i ∈ M n i × m i ( Z ) is an incidence matrix of H i , then " A n × m n × m A ∈ M ( n + n ) × ( m + m ) ( Z )is an incidence matrix of H (cid:16) H ; recall that n × m denotes the n × m all-one matrix.Both disjoint unions and complete unions naturally extend to families of more thantwo hypergraphs. These two operations are related by reflections of hypergraphs via theidentity ( H ⊕ H ) c = H c1 (cid:16) H c2 . (Multi-)graphs. A multigraph is a hypergraph Γ = ( V, E, | · | ) all of whose hyperedgesare edges. A graph is a multigraph without parallel edges; note that we allow graphsto contain loops. Two vertices v and v of a graph Γ are adjacent if there exists anedge e ∈ E with | e | = { v, v } ; we write v ∼ Γ v or v ∼ Γ v in that case. (This notation isunambiguous whenever V ∩ E = ∅ which we tacitly assume.) The set of neighbours of v ∈ V in Γ is { w ∈ V : v ∼ w } . A graph is simple if it contains no loops. The join Γ ∨ Γ of two simple graphs Γ and Γ is the simple graph obtained from the disjointunion Γ ⊕ Γ by adding an edge between each pair of vertices ( v , v ) ∈ V × V . Parameterising hypergraphs.
Up to isomorphism, a hypergraph H = ( V, E, | · | ) deter-mines and is determined by the cardinalities of the fibres µ I := { e ∈ E : | e | = I } ∈ N for I ⊂ V( H ). More formally: Definition 3.1.
Let V be a finite set. Given a vector µ = ( µ I ) I ⊂ V ∈ N P ( V )0 of non-negative multiplicities, define a hypergraph H ( µ ) withV( H ( µ )) = V, E( H ( µ )) = { ( I, j ) : I ⊂ V, j ∈ [ µ I ] } , | ( I, j ) | H ( µ ) = I. Modules and module representations from (hyper)graphs
In other words, H ( µ ) contains precisely µ I hyperedges with support I for each I ⊂ V .We often write m = P I ⊂ V µ I for the total number of hyperedges of H ( µ ). Clearly, foreach hypergraph H , there exists a unique vector µ as in Definition 3.1 such that H and H ( µ ) are isomorphic by means of an isomorphism fixing each vertex.We will use the following shorthand notation for the hypergraphs H ( µ ). For a finiteset V , suppose that we are given numbers µ I ∈ N for some but perhaps not all subsets I ⊂ V . We may then extend the collection of these µ I to a family µ as in Definition 3.1by setting µ J = 0 for the previously missing subsets J ⊂ V . We set H (cid:16) V (cid:12)(cid:12)(cid:12) X I µ I I (cid:17) := H ( µ );to further simplify our notation, we often drop coefficients µ I = 1 and summands µ I I with µ I = 0 from the left-hand side. Important families of (hyper)graphs.
The following hypergraphs will feature in severalplaces throughout this article; most have vertex set V = [ n ].The discrete ( hyper ) graph on n vertices (often called an “empty graph” in the litera-ture) is ∆ n := H ([ n ] |
0) := H ([ n ] | n ]) . (3.1)The n × m block hypergraph is the hypergraph BH n,m := H ([ n ] | m [ n ]) . (3.2)We denote the reflection of BH n,m by PH n,m ; that is, PH n,m = H ([ n ] | m ∅ ). Moregenerally, given n = ( n , . . . , n r ) ∈ N r and m = ( m , . . . , m r ) ∈ N r , let BH n , m := BH n ,m ⊕ · · · ⊕ BH n r ,m r and (3.3) PH n , m := BH c n , m = PH n ,m (cid:16) · · · (cid:16) PH n r ,m r . (3.4)The complete graph on n vertices isK n := H ([ n ] | X (cid:54) i The cycle graph on n (cid:62) n := H ([ n ] | X (cid:54) i RV, RE ) , ( v, e ) [ ve ];write η = η Z . We refer to η as the (absolute) incidence representation of H . Note thatthe notation η R is unambiguous: if R → S is a ring map, then ( η R ) S = η S . Description of η in terms of matrices. Write m = | E | , n = | V | , V = { v , . . . , v n } , and E = { e , . . . , e m } . Let M = n [ a ij ] ∈ M n × m ( R ) : a ij = 0 whenever v i 6∈ | e j | o . Then η R , as defined above, is isotopic (see §2.1) to the inclusion M , → M n × m ( R ). Incidence modules. Let X = ( X v ) v ∈ V consist of algebraically independent variablesover R . Define inc ( H ; R ) := D X v e : v ∼ H e ( v ∈ V, e ∈ E ) E (cid:54) R [ X ] E. The incidence module of H over R isInc( H ; R ) := R [ X ] E inc ( H ; R ) ;the (absolute) incidence module of H is Inc( H ) := Inc( H ; Z ). Clearly,Inc( H ; R ) ≈ R [ X ] M e ∈ E R [ X ] h X v : v ∈ | e |i . (3.10) Lemma 3.2. Inc( H ; S ) = Inc( H ; R ) S [ X ] for each ring map R → S . Modules and module representations from (hyper)graphs Proof. Immediate from the right exactness of tensor products. (cid:7) The incidence module of H determines the ask zeta functions associated with η : Proposition 3.3. For each compact DVR O , ζ ask η O ( s ) = (1 − q − ) − ζ Inc( H ; O ) ( s − | V | + | E | ) . Proof. By Lemma 2.3, Im( C I ( H ) ,V,Eη O ( X )) = inc ( H ; O ) (see (2.1)) whence Inc( H ; O ) =Coker (cid:16) C I ( H ) ,V,Eη O ( X ) (cid:17) . The claim thus follows from Corollary 2.13. (cid:7) For any suitable ring R , we refer to ζ ask η R ( s ) as the ask zeta function of H over R . Wecan use the structure of Inc( H ; R ) in (3.10) to make Proposition 3.3 more explicit. Proposition 3.4. For each compact DVR O , ζ ask η O ( s ) = (1 − q − ) − Z O V × O | y | s −| V | + | E |− Y e ∈ E k x e ; y k − d µ O V × O ( x, y ) , where x e := { x v : v ∈ | e |} .Proof. For ideals a , b / R of a ring R , there is a natural R -module isomorphism R/ a ⊗ R R/ b ≈ R/ ( a + b ); this follows e.g. from [10, Ch. I, §2, no. 8]. Equivalently, given asurjective ring map R λ −→ S , we have R/ a ⊗ R S ≈ S S/ ( a λ ). Let x ∈ O V and 0 = y ∈ O .Using (3.10) and Remark 2.11, we then obtain O -module isomorphismsInc( H ; O ) ⊗ O [ X ] ( O /y ) x ≈ M e ∈ E O [ X ] h X v : v ∈ | e |i ⊗ O [ X ] ( O /y ) x ≈ M e ∈ E O h x e ; y i . The common cardinality of these modules is therefore Q e ∈ E k x e ; y k − . The claim nowfollows from Corollary 2.13. (cid:7) Remark 3.5. Suppose that H = BH n,m is the n × m block hypergraph; see (3.2). In termsof matrices, η R parameterises all of M n × m ( R ). A formula for the ask zeta function of (theidentity on) M n × m ( O ) is given in [57, Proposition 1.5]; see Example 5.10(i). The integralin the proof of this proposition is exactly the corresponding special case of Proposition 3.4here. Similarly, the determination of the ask zeta functions of modules of (strictly) uppertriangular matrices over O in [57, Proposition 5.15] proceeded by computing the integralin the corresponding special case of Proposition 3.4; see Example 5.10(ii). We thusrecognise various ad hoc arguments from [57] as instances of the cokernel formalism here.As we will see in §5, Proposition 3.4 can be used to produce an explicit formula forthe ask zeta functions associated with all hypergraphs on a given vertex set.29 Modules and module representations from (hyper)graphs Let Γ = ( V, E, | · | ) be a graph. We construct two module representations γ + and γ − associated with Γ which we call the adjacency representations of Γ. The first of thesemodule representations is defined for all graphs Γ and parameterises symmetric matriceswith suitably constrained support. The second is defined whenever Γ is simple andparameterises antisymmetric matrices with support constrained by Γ. For our purposes,the antisymmetric case is usually more interesting.Denote the exterior (resp. symmetric) square of a module M by M ∧ M (resp. M (cid:12) M ). Alternating case: construction of γ − . Let Γ be simple. For an R -module M , let( M ∧ M ) ∗ so M −−→ Hom( M, M ∗ )be the module representation which sends ψ ∈ ( M ∧ M ) ∗ to the map M → M ∗ , m ( n ( m ∧ n ) ψ ) . If M = R n , then the evident choices of bases furnish an isotopy between so M and theinclusion so n ( R ) , → M n ( R ), of alternating (= antisymmetric with zero diagonal) n × n matrices into M n ( R ). We define the negative adjacency representation γ R − of Γ over R to be the composite (cid:18) RV ∧ RV N (Γ , − R ) (cid:19) ∗ (cid:26) ( RV ∧ RV ) ∗ so RV −−−→ Hom( RV, ( RV ) ∗ ) , where N (Γ , − R ) is the submodule of RV ∧ RV generated by all v ∧ w such that v, w ∈ V are non-adjacent in Γ and the first map is the dual of the quotient map. For an explicitdescription of γ R − in terms of matrices, let V = { v , . . . , v n } , where n = | V | . Let M − = n [ a ij ] ∈ so n ( R ) : a ij = 0 whenever v i v j o ;cf. the definition of M − (Γ) in §1.4. It is easy to see that γ R − is isotopic to the inclusion M − , → M n ( R ); a proof is implicitly given in the proof of Proposition 3.7 below.We refer to γ − := γ Z − as the (absolute) negative adjacency representation of Γ. As withincidence representations in §3.2, this notation is unambiguous: ( γ R − ) S = γ S − for eachring map R → S . Symmetric case: construction of γ + . Let Γ be not necessarily simple. For an R -module M , let ( M (cid:12) M ) ∗ Sym M −−−−→ Hom( M, M ∗ )be the module representation which sends ψ ∈ ( M (cid:12) M ) ∗ to the map M → M ∗ , m ( n ( m (cid:12) n ) ψ ) . Sym R n is isotopic to the inclusion Sym n ( R ) , → M n ( R ) of symmetric matrices.30 Modules and module representations from (hyper)graphs We define the positive adjacency representation γ R + of Γ over R to be the composite (cid:18) RV (cid:12) RV N (Γ , +1; R ) (cid:19) ∗ (cid:26) ( RV (cid:12) RV ) ∗ Sym RV −−−−→ Hom( RV, ( RV ) ∗ ) , where N (Γ , +1; R ) is the submodule of RV (cid:12) RV generated by all v (cid:12) w such that v, w ∈ V are non-adjacent in Γ. In terms of matrices, let V = { v , . . . , v n } , where n = | V | . Let M + = n [ a ij ] ∈ Sym n ( R ) : a ij = 0 whenever v i v j o ;cf. §1.1 and the definition of M + (Γ) in §1.4. Then γ R + is isotopic to the inclusion M + , → M n ( R ). As above, we call γ + := γ Z + the (absolute) positive adjacency representation of Γ. Adjacency modules. Let X = ( X v ) v ∈ V as in §3.2. For v, w ∈ V , define[ v, w ; ± 1] := ( X v w ± X w v, if v = w, ± X v v, if v = w, an element of Z [ X ] V , and adj (Γ , ± R ) := D [ v, w ; ± 1] : v, w ∈ V, v ∼ w E (cid:54) R [ X ] V. The (positive resp. negative) adjacency module of Γ over R isAdj(Γ , ± R ) := R [ X ] V adj (Γ , ± R ) . Lemma 3.6. Adj(Γ , ± S ) = Adj(Γ , ± R ) S [ X ] for each ring map R → S . (cid:7) We write Adj(Γ , ± 1) := Adj(Γ , ± Z ). Adjacency modules and ask zeta functions of graphs. In the same way that incidencemodules of hypergraphs determine the ask zeta functions associated with incidencerepresentations, adjacency modules of Γ are related to ask zeta functions derived from γ ± . Proposition 3.7. For each compact DVR O , ζ ask γ O ± ( s ) = (1 − q − ) − ζ Adj(Γ , ± O ) ( s ) . (Here, we assume that Γ is simple in the negative case.)Proof. We only spell out the “negative case”; the positive one can be established alongsimilar lines. Let v be an arbitrary total order on V . Let A (Γ , v ) := { ( v, w ) ∈ V × W : v ∼ w and v v w } . Let V ∗ = { v ∗ : v ∈ V } denote the dual basis associated with the basis V of RV .The images of the symbols v ∧ w for ( v, w ) ∈ A (Γ , v ) form a basis of RV ∧ RV N (Γ , − R ) ; let31 Modules and module representations from (hyper)graphs Φ := { φ vw : ( v, w ) ∈ A (Γ , v ) } denote the associated dual basis of (cid:16) RV ∧ RV N (Γ , − R ) (cid:17) ∗ , indexedin the natural way. Define a module representation R A (Γ , v ) θ = θ R v −−−→ Hom( RV, RV ),where for ( v, w ) ∈ A (Γ , v ) and u ∈ V , u (( v, w ) θ ) = + w, if u = v, − v, if u = w, , otherwise . It follows that the diagram RV ( v,w ) θ −−−−→ RV (cid:13)(cid:13)(cid:13) y ν RV φ vw γ R − −−−−→ ( RV ) ∗ commutes, where RV ν −→ ( RV ) ∗ is the isomorphism vν = v ∗ ( v ∈ V ). Hence, γ R − and θ are isotopic (see §2.1). Lemma 2.3 shows thatIm (cid:16) C A (Γ , v ) ,V,Vθ ( X ) (cid:17) = D X v w − X w v : ( v, w ) ∈ A (Γ , v ) E = adj (Γ , − R )whence Coker (cid:16) C Φ ,V,V ∗ γ R − − ( X ) (cid:17) ≈ R [ X ] Coker (cid:16) C A (Γ , v ) ,V,Vθ ( X ) (cid:17) = Adj(Γ , − R ) . The claim now follows from Corollary 2.13. (cid:7) Let Γ = ( V, E, | · | ) be a simple graph. Let v be an arbitrary total order on V . Define analternating bilinear map (cid:5) : Z V × Z V → Z E by letting, for v, w ∈ V with v (cid:64) w , v (cid:5) w := ( e, if there exists e ∈ E with | e | = { v, w } , , otherwise . We leave it to the reader to verify that the isomorphism type of the Baer groupscheme G (cid:5) (see §2.4) associated with (cid:5) only depends on Γ and not on the chosen totalorder v . We call G Γ := G (cid:5) the graphical group scheme associated with Γ. If Γ isa cograph (see §§1.7, 7.1), we talk about the cographical group scheme associatedwith Γ. By a ( co -) graphical group (over R ) we mean a group of rational points ofa (co-)graphical group scheme, i.e. a group of the form G Γ ( R ) for some ring R and(co)graph Γ. 32 Modules and module representations from (hyper)graphs Remark 3.8. The group G Γ ( Z ) of Z -points of G Γ is a finitely generated torsion-freenilpotent group. It admits the presentation G Γ ( Z ) ≈ D V t E (cid:12)(cid:12)(cid:12) [ v, w ] = e for e ∈ E with | e | = { v, w } and v (cid:64) w, [ v, w ] = 1 for non-adjacent v, w ∈ V , and[ v, e ] = [ e, f ] = 1 for all v ∈ V and e, f ∈ E E . Equivalently, G Γ ( Z ) is the maximal nilpotent quotient of nilpotency class at most 2 ofthe right-angled Artin group h V | [ v, w ] = 1 for all non-adjacent v, w ∈ V i associated with the complement of Γ; see e.g. [18]. It will prove advantageous for ourgraph-theoretic arguments in §§6–7 to work with Γ rather than with its complement.The following variant of Proposition 1.1 (which was proved in §2.4) will be crucial inestablishing Corollary B in §6.1. Proposition 3.9. Let Γ be a simple graph with m edges and let γ − denote its negativeadjacency representation over Z . Let R be the ring of integers of a local or global field ofarbitrary characteristic. Then ζ cc G Γ ⊗ R ( s ) = ζ ask γ R − ( s − m ) .Proof. Let Z V α −→ Hom( Z V, Z E ) be the module representation with v ( wα ) = v (cid:5) w forall v, w ∈ V . By Proposition 2.5, ζ cc G Γ ⊗ R ( s ) = ζ ask α R ( s − m ). A straightforward calculationas in the proof of Proposition 3.7 shows that the dual α • (see §2.1) of α is isotopic to γ − .Using Theorem 2.4 and the final argument from the proof of Proposition 1.1 (see §2.4),we conclude that ζ ask α R ( s ) = ζ ask γ R − ( s ) which completes the proof. (cid:7) Disjoint unions and joins. Let Γ and Γ be simple graphs. Recall from §3.1 that Γ ⊕ Γ and Γ ∨ Γ denote the disjoint union and join of Γ and Γ , respectively. Clearly, G Γ ⊕ Γ and G Γ × G Γ are isomorphic group schemes whence G Γ ⊕ Γ ( R ) ≈ G Γ ( R ) × G Γ ( R ) foreach ring R . Denote the lower central series of a group G by G = γγγ ( G ) (cid:62) γγγ ( G ) (cid:62) · · · .For groups G and G , let G (cid:16) G := ( G ∗ G ) /γγγ ( G ∗ G ) (3.11)be their free class- -nilpotent product , i.e. the maximal nilpotent quotient of class atmost 2 of the free product G ∗ G . Note that G Γ ∨ Γ ( R ) ≈ G Γ ( R ) (cid:16) G Γ ( R ) if R = Z or, more generally, R = Z /n Z for n ∈ Z .In particular, we conclude that the class of cographical groups over Z is precisely thesmallest class of torsion-free finitely generated groups which contains Z and which isclosed under taking both direct and free class-2-nilpotent products.33 Modules over toric rings and associated zeta functions By Corollary 2.13, the functions ζ M ( s ) attached to modules M over polynomial ringsgeneralise ask zeta functions. In this section, we introduce a further generalisation ofthese functions by replacing polynomial rings by more general toric rings. This moregeneral setting will provide us with a sufficient criterion (Proposition 4.8) for provinguniformity results such as Theorem A. Part (i) of the latter will be proved here whileparts (ii)–(ii) will be proved in §6.Throughout, as before, V is a finite set and R is a ring. We recall some standard notions from convex and toric geometry; see [21, Ch. 1–3].Unless otherwise indicated, by a cone in R V we mean a closed, rational, and polyhedralcone—in other words, cones are finite intersections of Z -defined linear half-spaces in R V .A fan in R V is a non-empty finite set F consisting of cones in R V such that(i) every face of every cone in F belongs to F and(ii) the intersection of any two cones in F is a common face of both.The support of a fan F is |F | = S F . The fan F is complete if |F | = R V . Let F and G be two fans in R V . We say that G refines F if every cone in G is containedin some cone in F . The coarsest common refinement of F and G is the fan (!) F ∧ G := { σ ∩ τ : σ ∈ F , τ ∈ G} ; its support is |F ∧ G| = |F | ∩ |G| .Let · be the standard inner product x · y = P v ∈ V x v y v on R V . If σ ⊂ R V is a cone,then so is its dual σ ∗ = { x ∈ R V : ∀ y ∈ σ : x · y (cid:62) } .Let σ ⊂ R (cid:62) V be a cone. Recall that a preorder on a set is a reflexive and transitiverelation. If all elements are comparable, then the preorder is total . We note that“total preorders” and “weak orders” (cf. §§1.6, 5.1) are equivalent concepts. We define apreorder (cid:54) σ on Z V by letting x (cid:54) σ y if and only if y − x ∈ σ ∗ . Lemma 4.1. For every fan F in R V and finite set Φ ⊂ Z V , there exists a refinement F of F with |F | = |F | and such that (cid:54) σ induces a total preorder on Φ for each σ ∈ F .Proof. We may assume that Φ = ∅ . For x ∈ R V , let x ± := { y ∈ R V : ± x · y (cid:62) } bethe associated linear half-space and x = := x + ∩ x − = x ⊥ . We obtain a complete fan F x := { x + , x − , x = } consisting of precisely three cones, except when x = 0 in which case F x = { R V } . Clearly, the refinement F := F ∧ V x,y ∈ Φ F x − y has the desired property. (cid:7) Toric rings and affine toric schemes. Let σ ⊂ R V be a cone. By Gordan’s lemma(see [21, Proposition 1.2.17]), the additive monoid σ ∗ ∩ Z V is finitely generated. Let X = ( X v ) v ∈ V consist of algebraically independent variables over R . For α ∈ Z V , write34 Modules over toric rings and associated zeta functions X α = Q v ∈ V X α v v . In the same way, we define x α , where x = P v ∈ V x v v and all the x v areunits (in some ambient ring). We let R σ := R [ X α : α ∈ σ ∗ ∩ Z V ]be the toric ring associated with σ and R . We let X σ,R = Spec( R σ ) be the associated affine toric scheme over R ; we write X σ := X σ, Z . Rational points over DVRs. Let σ ⊂ R (cid:62) V be a cone and let O be a DVR . Recall that ν denotes the normalised valuation on O . For x = P v ∈ V x v v ∈ O V with Q v ∈ V x v = 0, wewrite ν ( x ) := P v ∈ V ν ( x v ) v ∈ Z V . Define σ ( O ) := n x ∈ O V : Y v ∈ V x v = 0 and ν ( x ) ∈ σ o . Alternatively, σ ( O ) admits the following dual description. Lemma 4.2. σ ( O ) = { x ∈ O V : Y v ∈ V x v = 0 and x α ∈ O for each α ∈ σ ∗ ∩ Z V } .Proof. Let x ∈ O V with Q v ∈ V x v = 0. Then x α ∈ O if and only if ν ( x α ) = ν ( x ) · α (cid:62) α ∈ σ ∗ ∩ Z V if and only if ν ( x ) ∈ σ ∗∗ = σ . (cid:7) Recall that X σ = Spec( Z σ ). As before, write X = ( X v ) v ∈ V . Lemma 4.3. Let ϕ be the natural map X σ ( O ) → O V induced by the inclusion σ ⊂ R (cid:62) V .Let Z = { x ∈ O V : Q v ∈ V x v = 0 } and let Z be the preimage of Z under ϕ . Then ϕ inducesa bijection X σ ( O ) \ Z → σ ( O ) .Proof. Let x ∈ X σ ( O ) \ Z . Then x corresponds to a ring map Z σ λ −→ O . Let x := x ϕ sothat x v := X v λ . Since x Z , we have Q v ∈ V x v = 0. Let α ∈ σ ∗ ∩ Z V be arbitrary. Thenthere exists β ∈ Z (cid:62) V ⊂ σ ∗ with α + β ∈ Z (cid:62) V . We conclude that x α + β = ( X α + β ) λ =( X α ) λ · ( X β ) λ = ( X α ) λ · x β and therefore ( X α ) λ = x α ∈ O \ { } . By Lemma 4.2, x = x ϕ ∈ σ ( O ). We have thus shown that λ (hence x ) is uniquely determined by x which implies that ϕ injectively maps X σ ( O ) \ Z onto a subset of σ ( O ). It remains toshow that the latter subset is all of σ ( O ). Indeed, for each y ∈ σ ( O ), by Lemma 4.2, weobtain a ring map Z σ → O , X α y α whose corresponding point y ∈ X σ ( O ) does not belong to Z and satisfies y ϕ = y . (cid:7) We henceforth tacitly embed σ ( O ) ⊂ X σ ( O ) via Lemma 4.3.35 Modules over toric rings and associated zeta functions We now generalise the definition of the zeta functions ζ M (see §2.6) attached to modulesover polynomial rings to those over toric rings.Let σ ⊂ R (cid:62) V be a cone. Recall that X σ = Spec( Z σ ) is the affine toric scheme (over Z )associated with σ . Let O be a compact DVR and let M be a finitely generated O σ -module. Generalising the definition of M x in §2.6 (cf. Lemma 2.10), for each x ∈ X σ ( O )(= X σ, O ( O )), let M x denote the O -module M ⊗ O σ O , where the O σ -module structure on O is induced by the ring map O σ → O corresponding to x . When σ = R (cid:62) V , we recoverthe definition of M x given in §2.6. Recall that we identify σ ( O ) ⊂ O V with a subset of X σ ( O ) via Lemma 4.3. Definition 4.4. Define a zeta function ζ M ( s ) := Z σ ( O ) × O | y | s − · | M x ⊗ O /y | d µ O V × O ( x, y ) . Remark 4.5. (i) If M is an O [ X ]-module (= O R (cid:62) V -module), then we recover Definition 2.9.(ii) The function ζ M ( s ) only depends on the isomorphism type of M as an O σ -module.(iii) Exactly as in Remark 2.11, we may identify M x ⊗ O O /y = M ⊗ O σ ( O /y ) x . Lemma 4.6. Let o ⊂ R (cid:62) V be a cone. Let M o be a finitely generated O o -module. Let F be a fan with |F | = o . For σ ∈ F , let M σ denote the O σ -module M o ⊗ O o O σ . Then ζ M o ( s ) = X ∅ =Σ ⊂F ( − | Σ | +1 ζ M σ ( s ) , where we wrote σ := T Σ .Proof. This follows by combining the inclusion-exclusion principle and the identification( M σ ) x = ( M o ) x for σ ∈ F and x ∈ σ ( O ) ⊂ o ( O ) (“transitivity of base change”). (cid:7) Global setting. We now provide a global setting for the functions ζ M . Let R be aNoetherian ring, let o ⊂ R (cid:62) V be a cone, and let M o be a finitely generated R o -module.For each ring map R λ −→ O , let ζ M o ,λ ( s ) := ζ M o ⊗ Ro O o ( s ) , where the ring map R o λ o −→ O o is induced by λ ; when the reference to λ is clear, we alsowrite ζ M o , O in place of ζ M o ,λ in the following.This global setting is compatible with Lemma 4.6 in the sense that for a cone σ ⊂ o , bytransitivity of base change, we may identify ( M o ⊗ R o O o ) ⊗ O o O σ = ( M o ⊗ R o R σ ) ⊗ R σ O σ .36 Modules over toric rings and associated zeta functions Toric properties. Let R be a ring. Let P be a property of objects of types A , B , . . . (e.g. modules) defined over all base R -algebras of the form R σ for cones σ containedwithin some ambient cone in R V . We assume that (i) P is invariant under isomorphisms,(ii) for each inclusion τ ⊂ σ of such cones, every R σ -object A gives rise to an R τ -object A ⊗ R σ R τ , and (iii) this base change operation is transitive (up to isomorphism). We saythat specific objects A, B, . . . over a specific R -algebra R σ have the property P torically if there exists a fan F of cones in R V with |F | = σ such that the R τ -objects A ⊗ R σ R τ , B ⊗ R σ R τ , . . . have property P for all τ ∈ F . Transitivity of toric properties. Let the property P be as above. Further suppose that P is stable under shrinking cones in the sense that for each inclusion τ ⊂ σ of cones,whenever objects A, B, . . . over R σ have P , then so do A ⊗ R σ R τ , B ⊗ R σ R τ , . . . .Let o ⊂ R V be a cone and let F be a fan of cones in R V with support o . Let A, B, . . . be objects over R o and suppose that the R σ -objects A ⊗ R o R σ , B ⊗ R o R σ , . . . toricallyhave property P for each σ ∈ F . Then A, B, . . . themselves torically have property P over R o ; for a proof, apply §6.3 (which is self-contained) below; cf. Corollary 6.16. Combinatorial modules. Let σ ⊂ R V be a cone. Let R be a ring. By a monomialideal I of R σ , we mean an ideal generated by (finitely many) Laurent monomials X α for α ∈ σ ∗ ∩ Z V . We say that an R σ -module is combinatorial if it is isomorphic to R σ /I ⊕ · · · ⊕ R σ /I m , where each I j is a monomial ideal of R σ . Example 4.7 (Incidence modules are combinatorial) . Let H be a hypergraph with vertexset V . By (3.10), the incidence module Inc( H ; R ) is a combinatorial R [ X ]-module, where X = ( X v ) v ∈ V . Proposition 4.8 (Uniformity of zeta functions of torically combinatorial modules) . Let σ ⊂ R (cid:62) V be a cone and let M be a torically combinatorial R σ -module. Then thereexists W ( X, T ) ∈ Q ( X, T ) such that ζ M,λ ( s ) = W ( q, q − s ) for each compact DVR O andring map R λ −→ O .Proof. Fix R λ −→ O . First, let A ⊂ σ ∗ ∩ Z V be a finite set. Let I = h X α : α ∈ A i / R σ and N = R σ /I . Let x ∈ σ ( O ) and y ∈ O \ { } . The evident free presentation R σ A → R σ → N → O /y -module( N ⊗ R σ O σ ) x ⊗ O O /y ≈ N ⊗ R σ ( O /y ) x ≈ O / h x α ( α ∈ A ); y i =: N x,y ;cf. Proposition 3.4. In particular, | N x,y | = k x α ( α ∈ A ); y k − , independently of λ .Next, by Lemma 4.6, after shrinking σ if necessary, we may assume that M is in factcombinatorial instead of merely torically combinatorial, say M ≈ R σ /I ⊕ · · · ⊕ R σ /I m ,where I j = h X α : α ∈ A j i / R σ and each A j ⊂ σ ∗ ∩ Z V is finite. We thus obtain ζ M,λ ( s ) = Z σ ( O ) × O | y | s − m Q j =1 k x α ( α ∈ A j ); y k d µ O V × O ( x, y ) . Ask zeta functions of hypergraphs The claimed uniformity result for p -adic integrals defined by such monomial expressionsis well-known, see e.g. [54, Proposition 3.9]. (cid:7) Remark 4.9. (i) If R admits any ring map to any compact DVR , then the rational function W ( X, T )in Proposition 4.8 is uniquely determined. Indeed, if R → O is such a ring map,where O has residue field size q , then we obtain ring maps from R to a compact DVR with residue field size q f for each f (cid:62) 1. Uniqueness of W ( X, T ) then essentiallyboils down to the fact that infinite subsets of C are Zariski dense.(ii) The preceding condition is satisfied, in particular, if R is finitely generated over Z .To see that, let m be an arbitrary maximal ideal of R . By the Nullstellensatz forJacobson rings (see e.g. [27, Theorem 4.19]), R/ m is then a finite field. We thene.g. obtain a ring map R → ( R/ m )[[ z ]]. Proof of Theorem A(i). Combine Example 4.7 and Proposition 4.8. (cid:7) In §6, we will show that negative adjacency modules of graphs are always toricallycombinatorial and that their positive counterparts are torically combinatorial over anyground ring in which 2 is invertible. Let H = ( V, E, | · | ) be a hypergraph. We write n = | V | and and m = | E | . As explainedin §3.2, this allows us to think of the incidence representation η of H in terms of ageneric n × m matrix with support constrained by the hyperedge support function | · | .In §5.1, we derive an explicit combinatorial formula for the rational function W H ( X, T )in Theorem A(i) and thus, for each compact DVR O , for the ask zeta function ζ ask η O ( s ).We then consider two natural operations of hypergraphs: disjoint unions (see §5.2) andcomplete unions (see §5.3 and §3.1). As special cases, we derive explicit formulae for askzeta functions of hypergraphs with pairwise disjoint (resp. codisjoint) hyperedge supportsin §5.2.1 (resp. §5.3.1). In §5.4, we describe the effects of four further fundamentalhypergraph operations on the rational functions W H ( X, T ). In §5.5, we use our explicitformulae to deduce crucial analytic properties of local and global ask zeta functionsassociated with hypergraphs.Throughout this section and beyond, we use the notation ( d ) = 1 − q − d . The main result of this section, Corollary 5.6, provides an explicit formula for the rationalfunction W H ( µ ) ( X, T ) (see Theorem A(i)) associated with an arbitrary hypergraph H ( µ )given by a vector µ of hyperedge multiplicities; see Definition 3.1. This formula will, inparticular, imply Theorem C. 38 Ask zeta functions of hypergraphs Socles. For applications later on, it will prove advantageous to study the rationalfunction W H ( µ ) ( X, T ) in (what appears to be) a slightly more general setup. Definition 5.1. Given a d -element set D with V ∩ D = ∅ and a vector µ of hyperedgemultiplicities as in Definition 3.1, define a hypergraph H ( µ , D ) with vertex set D t V and vector of hyperedge multiplicities ( ν J ) J ⊂ D t V given by ν J = ( µ I , if J = D t I for some I ⊂ V, , otherwise.Informally speaking, the hypergraph H ( µ , D ) arises from H ( µ ) by inflating eachhyperedge by the same fixed set (“socle”) D . Thus, if A ∈ M n × m ( Z ) is an incidencematrix of H ( µ ), then " d × m A ∈ M ( d + n ) × m ( Z )is an incidence matrix of H ( µ , D ).We now derive an explicit formula for W H ( µ ,D ) ( X, T ); the shape of this formula willoften allow us to reduce to the case D = ∅ . Setup and strategy. From now on, let O be an arbitrary compact DVR with residue fieldcardinality q . Without loss of generality, suppose that 0 V t D and write D := D t { } .Recall from §1.11 that for a non-trivial O -module M , we write M × = M \ P M and { } × = { } . For J ⊂ V , define p -adic integrals Z J,D ( s ) := Z J,D (cid:16) s , ( s I ) I ⊂ J ) (cid:17) := Z O J × O D | y | s Y I ⊂ J k x I ; y k s I d µ O J × O D ( x, y ) and I J,D ( s ) := Z ( O J ) × × P D | y | s Y I ⊂ J k x I ; y k s I d µ O J × D ( x, y ) , (5.1)where x I := ( x i : i ∈ I ) ∈ O I ⊂ O J . Depending on context, we regard Z J,D ( s ) and I J,D ( s ) both as functions of the 1 + 2 | J | variables s and ( s I ) I ⊂ J and also as functions ofthe 1 + 2 | V | variables s and ( s I ) I ⊂ V ; in any case, s and s ∅ are different variables.Let η µ ,D be the incidence representation of H ( µ , D ); see §3.2. We seek to determine W H ( µ ,D ) ( X, T ) (and hence also W H ( µ ) ( X, T )) by expressing ζ ask η O µ ,D ( s ) as a rational functionin q and q − s . By Proposition 3.4, ζ ask η O µ ,D ( s ) = (1 − q − ) − Z O V × O D | y | s − ( n + d )+ m − Y I ⊂ V k x I ; y k − µ I d µ O V × O D ( x, y )= (1 − q − ) − Z V,D (cid:16) s − ( n + d ) + m − , ( − µ I ) I ⊂ V (cid:17) . (5.2)This allows us to study W H ( X, T ) by analysing the functions Z V,D ( s ).39 Ask zeta functions of hypergraphs A recursive formula. Our first goal is to derive a recursive formula for Z V,D ( s ); seeProposition 5.2. In the following, we write t I = q − s I (where I ⊂ V or I = 0) and t = q − s .We identify O D = O × O D and decompose O V × O D = O V × O × O D in the form (cid:0) ( O V ) × × P × P D (cid:1) t ( P V × P × P D ) t (cid:0) O V × O × × O D (cid:1) t (cid:0) O V × P × ( O D ) × (cid:1) . (5.3)Write gp ( x ) = x/ (1 − x ) and gp ( x ) = 1 / (1 − x ). Using a change of variables (cf.[37, Proposition 7.4.1]) and the well-known (and easily proved) identity Z P | y | s d µ O ( y ) = (1) gp (cid:16) q − t (cid:17) , (5.4)we rewrite Z V,D ( s ) as Z V,D ( s ) = I V,D ( s ) + q − n − − d t Y I ⊂ V t I ! Z V,D ( s ) + (1) (cid:16) d )gp (cid:16) q − t (cid:17)(cid:17) . For J ⊂ V , let Z J,D ( s ) := 1 − q − d − t − q − t + (1) − I J,D ( s ); (5.5)here, and in the following subsections, we often abbreviate Z J ( s ) := Z J, ∅ ( s ). Thus, Z V,D ( s ) = gp (cid:16) q − n − − d t Y I ⊂ V t I (cid:17) (1) Z V,D ( s ) (5.6)and hence, by combining (5.2) and (5.6), ζ ask η O µ ,D ( s ) = 11 − t Z V,D (cid:16) s − ( n + d ) + m − , ( − µ I ) I ⊂ V (cid:17) . (5.7)The function Z V,D ( s ) admits the following recursive expression. Proposition 5.2. Z V,D ( s ) = 1 − q − d − t − q − t + X J (cid:40) V (1) n −| J | gp (cid:16) q − d − −| J | t Y I ⊂ J t I (cid:17) Z J,D ( s ) . (5.8) Proof. We decompose the first factor of the domain of integration of I V,D ( s ) definedin (5.1) according to precisely which entries of x ∈ ( O V ) × are P -adic units; no suchentry affects the integrand. By Fubini’s theorem, we may then split off the relevantcopies of O × , each of Haar measure (1), and write I V,D ( s ) = X J (cid:40) V (1) n −| J | Z P J × P D | y | s Y I ⊂ J k x I ; y k s I d µ O J × O D ( x, y ) | {z } =: I ◦ J,D ( s ) . (5.9)40 Ask zeta functions of hypergraphs A change of variables shows that I ◦ J,D ( s ) = q − d − −| J | t (cid:16) Y I ⊂ J t I (cid:17) Z O J × O D | y | s Y I ⊂ J k x I ; y k s I d µ O J × O D ( x, y ) . Using a decomposition of O J × O D = O J × O × O D analogous to (5.3), we obtain I ◦ J,D ( s ) q − d − −| J | t (cid:16) Q I ⊂ J t I (cid:17) = I J,D ( s ) + I ◦ J,D ( s ) + (1) − q − − d t − q − t ! and hence I ◦ J,D ( s ) = gp q − d − −| J | t Y I ⊂ J t I ! (1) − q − − d t − q − t ! + I J,D ( s ) ! = (1) gp q − d − −| J | t Y I ⊂ J t I ! Z J,D ( s ) . (5.10)By combining (5.5) and (5.9)–(5.10), we finally obtain Z V,D ( s ) = 1 − q − − d t − q − t + (1) − I V,D ( s )= 1 − q − − d t − q − t + X J (cid:40) V (1) n −| J |− I ◦ J,D ( s )= 1 − q − − d t − q − t + X J (cid:40) V (1) n −| J | gp (cid:16) q − d − −| J | t Y I ⊂ J t I (cid:17) Z J,D ( s ) . (cid:7) An explicit formula in terms of weak orders. Our next goal is to translate the recur-sive formula in Proposition 5.2 into an explicit form given by a sum over a suitablecombinatorial object. Definition 5.3. Let d WO( V ) be the poset of flags of subsets of V . That is, d WO( V )consists of elements of the form y = (cid:16) I (cid:40) I (cid:40) · · · (cid:40) I ‘ (cid:17) , where ‘ (cid:62) I i ⊂ V for i = 1 , . . . , ‘ . Note that we allow both I = ∅ and I ‘ = V butdo not require either condition to be satisfied. We define the rank of y ∈ d WO( V ) to berk( y ) = | sup( y ) | = sup {| I | : I ∈ y } ∈ N ;empty flags have rank 0. We denote by g WO( V ) the subposet of d WO( V ) consisting of allflags of non-empty subsets of V only. We often write d WO n and g WO n instead of d WO([ n ])and g WO([ n ]), respectively. 41 Ask zeta functions of hypergraphs Remark 5.4. (i) Clearly, rk( y ) = 0 if and only if y is either the empty flag or the singleton flag ( ∅ ).At the other extreme, rk( y ) = | V | = n if and only if V ∈ y . The latter conditionis satisfied for precisely half of the elements of g WO( V ). The fact that V ∈ y ispermitted for elements y of g WO( V ) marks the difference between the latter andthe poset WO n of weak orders of n objects; cf. e.g. [62, Section 2.3]. In particular, | d WO( V ) | = | g WO( V ) | = 2 | WO n | = 2 f n , where f n denotes the n th Fubini numberas in §1.6; cf. [30], [19, p. 228], and (1.4).(ii) The poset g WO( V ) is isomorphic to WO ( n ) in [17, Section 3.1].We obtain the following explicit formula for Z V,D ( s ). Theorem 5.5. Z V,D ( s ) = 1 − q − d − t − q − t X y ∈ g WO( V ) (1) rk( y ) Y J ∈ y gp (cid:16) q − − n + | J |− d t Y I ⊂ V \ J t I (cid:17) . Proof. Recursively apply Proposition 5.2 to the terms Z J,D ( s ) on the right-hand sideof (5.8). (cid:7) In particular, using (5.7), we obtain the following explicit formulae for the rationalfunction W H ( µ ,D ) ( X, T ) associated with the hypergraph H ( µ , D ). Corollary 5.6. W H ( µ ,D ) ( X, T )= 1 − X n − m T (1 − X d + n − m T )(1 − T ) X y ∈ g WO( V ) (1 − X − ) rk( y ) Y J ∈ y gp X | J |− P I ∩ J = ∅ µ I T ! (5.11)= 1 − X n − m T − X d + n − m T X y ∈ d WO( V ) (1 − X − ) rk( y ) Y J ∈ y gp X | J |− P I ∩ J = ∅ µ I T ! (5.12)= 1 − X n − m T − X d + n − m T W H ( µ ) ( X, T ) . (cid:7) Proof of Theorem C. Apply Corollary 5.6 with D = ∅ . (cid:7) Remark 5.7. (i) For n = 0, we recover the formula for the ask zeta function associated with theblock hypergraph BH d,m in (3.2); see Example 5.10(i).(ii) The rational functions in (5.11) and (5.12) are reminiscent of the “generalisedIgusa function” I wo ( n ) ( X ) associated with the all-one-vector ( n ) = (1 , . . . , 1) in[17, Definition 3.5]. The curious factors (1 − X − ) rk( y ) , however, set these two typesof combinatorially defined functions apart.42 Ask zeta functions of hypergraphs Example 5.8. We write out the formulae for the functions W H ( µ ) ( X, T ) = W H ( µ , ∅ ) ( X, T )given in (5.12) for n ∈ { , } . We identify V = [ n ] and set, for J ⊂ [ n ], gp J := gp J ( µ ) :=gp (cid:16) X | J |− P I ∩ J = ∅ µ I T (cid:17) . Write gp i := gp { i } and similarly gp ij ··· := gp ( { i, j, . . . } ).(i) ( n = 2) The ranks of the six flags in g WO are given as follows. y ∈ g WO ( ) ( { , } ) ( { } (cid:40) { , } ) ( { } (cid:40) { , } ) ( { } ) ( { } )rk( y ) 0 2 2 2 1 1Thus W H ( µ ) ( X, T ) =11 − T (cid:16) − X − ) gp (1 + gp + gp ) + (1 − X − ) (gp + gp ) (cid:17) , (5.13)where the relevant substitutions are given by the numerical data X | J |− P I ∩ J = ∅ µ I T = X − µ − µ − µ T, for J = { , } ,X − µ − µ T, for J = { } ,X − µ − µ T, for J = { } . (ii) ( n = 3) Here, | g WO | = 26 and W H ( µ ) ( X, T ) = 11 − T (cid:16) − X − ) gp (1 + gp + gp + gp +gp (1+gp +gp )+gp (1+gp +gp )+gp (1+gp +gp ))+ (1 − X − ) (gp (1 + gp + gp ) + gp (1 + gp + gp )+ gp (1 + gp + gp )) + (1 − X − ) (gp + gp + gp ) (cid:17) ;we omit the lengthy substitutions.It seems remarkable how slight the dependence of W H ( µ ,D ) ( X, T ) on the “socle” D is.The final equality in Corollary 5.6 often allows us to reduce to the case D = ∅ or,equivalently, to assume that no vertex of our hypergraph is incident to every hyperedge. Let m = ( m , . . . , m n ) ∈ N n +10 and write m = m + · · · + m n . Recall the definition ofthe staircase hypergraph ΣH m from (3.9). The upper block-triagonal “staircase matrix” M m = " δ j> P ι
Proposition 5.9. W ΣH m ( X, T ) = 11 − T n − Y j =0 − X − n − j − P ι>j m ι T − X n − j − P ι>j m ι T . (5.14) Proof. Combine Proposition 3.4 and [57, Lemma 5.6]. (cid:7) Proposition 5.9 generalises several previously known results. Example 5.10. (i) If m = . . . = m n − = 0 and m n = m , then ΣH m = BH n,m . Proposition 5.9 yields,in accordance with [57, Proposition 1.5], W ΣH m ( X, T ) = W BH n,m ( X, T ) = 11 − T n − Y j =0 − X − n − j − m T − X n − j − m T = 1 − X − m T (1 − T )(1 − X n − m T ) . (ii) If m = 0 and m = . . . = m n = 1, then M m = [ δ i (cid:54) j ] ∈ M n ( Z ). Proposition 5.9yields, in accordance with [57, Proposition 5.13(ii)], W ΣH m ( X, T ) = 11 − T n − Y j =0 − X − n − j − ( n − j ) T − X n − j − ( n − j ) T = (1 − X − T ) n (1 − T ) n +1 . In this section, we consider ask zeta functions associated with disjoint unions of hyper-graphs. As our main result, in §5.2.1, we record an explicit formula for ask zeta functionsattached to hypergraphs with pairwise disjoint (hyperedge) supports. Hadamard products. Recall that the Hadamard product of two generating functions F ( T ) = ∞ P k =0 a k T k and G ( T ) = ∞ P k =0 b k T k with coefficients in some common field is F ( T ) ? G ( T ) := ∞ X k =0 a k b k T k . If F ( T ) and G ( T ) are both rational, then so is F ( T ) ? G ( T ); see [65, Proposition 4.2.5].44 Ask zeta functions of hypergraphs Disjoint unions. Let H , . . . , H r be hypergraphs with pairwise disjoint vertex sets V , . . . , V r . Let H := H ⊕ · · · ⊕ H r be the disjoint union of H , . . . , H r as in §3.1. Proposition 5.11. W H ⊕···⊕ H r ( X, T ) = W H ( X, T ) ? · · · ? W H r ( X, T ) .Proof. Let η i and η be the incidence representations of H i and H , respectively. We mayidentify η = η ⊕· · ·⊕ η r (see §2.1). Now apply [60, Lemma 3.1]; cf. [57, Corollary 3.6]. (cid:7) We conclude that the set of rational functions W H ( X, T ) associated with hypergraphs(or, equivalently, the class of rational functions given by the right-hand side of (5.11)) isclosed under taking Hadamard products.It is natural to seek to exploit this closure property. Let H i ≈ H ( µ ( i ) ) for a vector µ ( i ) of hyperedge multiplicities as in Definition 3.1. By combining Corollary 5.6 andProposition 5.11, we obtain W H ⊕···⊕ H r ( X, T ) = r ? i =1 X y i ∈ d WO( V i ) (1 − X − ) rk( y i ) Y J ∈ y i gp (cid:16) X | J |− P I ∩ J = ∅ µ ( i ) I T (cid:17) . (5.15)The right-hand side of (5.15) falls short of being truly explicit due to the rathermysterious nature of Hadamard products. On the other hand, Corollary 5.6 provides anexplicit formula for W H ( X, T ) in terms of the hyperedge multiplicity vector µ ∈ N P ( V )0 ,where V := V t · · · t V r and µ I := P ri =1 δ I ⊂ V i µ ( i ) I ∩ V i for I ⊂ V . This approach, however,takes no advantage of the fact that H is a disjoint union. As we will now see, it turns outthat we can do much better at least when each H i is a block hypergraph as in (3.2). Let n = ( n , . . . , n r ) ∈ N r and m = ( m , . . . , m r ) ∈ N r ; write n = n + · · · + n r and m = m + · · · + m r . Let H := BH n , m be the disjoint union of the block hypergraphs H i := BH n i ,m i ; see (3.3). Let V i be the set of vertices of H i and V = V t · · · t V r be thatof H . Note that n i × m i ∈ M n i × m i ( Z ) is the (unique!) incidence matrix of H i . It followsthat diag ( n × m , . . . , n r × m r ) ∈ M n × m ( Z ) (5.16)is an incidence matrix of BH n , m . Note that, up to reordering of rows and columnswhere necessary, this is the general form of incidence matrices of hypergraphs withdisjoint supports (i.e. whenever µ I µ J = 0, then I = J or I ∩ J = ∅ ) and which alsosatisfy S µ I > I = V . (We will see that the latter condition imposes no real restrictions,nor would allowing some m i = 0 offer anything new; see Remark 5.24.)In Corollary 5.6, we obtained an expression for W H ( X, T ) as a sum over d WO( V ) ≈ d WO n .Our main result (Corollary 5.14) of this section provides an expression for W H ( X, T ) asa sum over d WO r . Apart from better reflecting the structure of the hypergraph H , inlight of the rapid growth of Fubini numbers in (1.4), our formula has a more favourablecomplexity if r (cid:28) n ; see also Remark 5.20.45 Ask zeta functions of hypergraphs Auxiliary functions. We consider the specialisation Z ⊕ n ( s ) := Z ⊕ n ( s ; s V , . . . , s V r ) := Z V (cid:16) s , (cid:16) δ ∃ i ∈ [ r ]: I = V i s I (cid:17) I ⊂ V (cid:17) (5.17)of the function Z V ( s ) = Z V, ∅ ( s ) from (5.5). In other words, Z ⊕ n ( s ) is obtained from Z V ( s ) by setting all variables s I corresponding to subsets I ⊂ V to zero, except for thosesubsets equal to one of the pairwise disjoint sets V i . From (5.7) (with D = ∅ ), we obtain ζ ask η O ⊕···⊕ η O r ( s ) = 11 − t Z ⊕ n ( s − n + m − − m , . . . , − m r ) . Given J ⊂ [ r ], write n J = P j ∈ J n j and n J = ( n j ) j ∈ J . Generalising (5.17), we define Z ⊕ n J ( s ) := Z V (cid:18) s , (cid:16) δ ∃ j ∈ J : I = V j s I (cid:17) I ⊂ V (cid:19) . A recursive formula. We obtain the following recursive formula for Z ⊕ n ( s ); the proof issimilar to that of Proposition 5.2 and hence omitted. Proposition 5.12. Z ⊕ n ( s ) = 1 + X J (cid:40) [ r ] (cid:16) Y k J ( n k ) (cid:17) gp q − − n J t Y j ∈ J t V j ! Z ⊕ n J ( s ) . (cid:7) An explicit formula. Just as Proposition 5.2 implies Theorem 5.5, we obtain thefollowing by unravelling the recursive formula in Proposition 5.12. Theorem 5.13. Z ⊕ n ( s ) = X y ∈ g WO r (cid:16) Y i ∈ sup( y ) ( n i ) (cid:17) Y J ∈ y gp q − − n + n J t Y j ∈ [ r ] \ J t V j ! . (cid:7) In particular, Theorem 5.13 allows us to produce the following explicit formula for therational function W BH n , m ( X, T ) associated with the disjoint union BH n , m = r L i =1 BH n i ,m i . Corollary 5.14. W BH n , m ( X, T ) = X y ∈ d WO r (cid:16) Y i ∈ sup( y ) (1 − X − n i ) (cid:17) Y J ∈ y gp X P j ∈ J n j − m j T ! . (cid:7) (5.18) Example 5.15. For r = 2, formula (5.18) for the ask zeta function associated with thedisjoint union of two block hypergraphs BH n i ,m i reads W BH ( n ,n , ( m ,m ( X, T ) =11 − T (cid:0) − X − n )gp (cid:0) X n − m T (cid:1) + (1 − X − n )gp (cid:0) X n − m T (cid:1) +(1 − X − n )(1 − X − n )gp (cid:16) X n + n − m − m T (cid:17) (cid:0) (cid:0) X n − m T (cid:1) + gp (cid:0) X n − m T (cid:1)(cid:1)(cid:17) . (5.19)46 Ask zeta functions of hypergraphs It is instructive to compare (5.19) and the general formula (5.13) for W H ( X, T ) in thespecial case that H is a hypergraph on two vertices. Let H and H be hypergraphs on disjoint sets V and V of vertices. Recall from §3.1the definition of the complete union H (cid:16) H of H and H , a hypergraph with vertex set V t V . In the main result of this section, Corollary 5.17, we express W H (cid:16) H ( X, T ) interms of W H ( X, T ) and W H ( X, T ). In §5.3.1, we also record an explicit formula for therational function W H ( X, T ) whenever H has pairwise codisjoint hyperedge supports; suchhypergraphs are precisely the reflections (see §3.1) of those considered in §5.2.1.Let H i have n i vertices and m i hyperedges; write n = n + n , and m = m + m .Let H , H , and H := H (cid:16) H be given by the multiplicity vectors µ (1) , µ (2) , and µ ,respectively; cf. Definition 3.1. For I ⊂ V := V t V , let I i := I ∩ V i so that µ I = δ I = V µ (1) I + δ I = V µ (2) I . An auxiliary function. Recall the definition of Z V ( s ) = Z V, ∅ ( s ) from (5.5) and considerthe specialisation Z (cid:16) ( V ,V ) ( s ) := Z (cid:16) ( V ,V ) ( s , s (1) , s (2) ) := Z V (cid:16) s , (cid:16) δ I = V s (1) I + δ I = V s (2) I (cid:17) I ⊂ V (cid:17) . In other words, Z (cid:16) ( V ,V ) ( s ) is obtained from Z V ( s ) by setting all variables s I for I ⊂ V to zero, except for those I that contain one of the disjoint sets V and V ; note that thevariable s V is substituted by s (1) V + s (2) V . Thus, Z (cid:16) ( V ,V ) ( s ) is a function of 1 + 2 n + 2 n complex variables s , s (1) = (cid:16) s (1) I (cid:17) I ⊂ V , and s (2) = (cid:16) s (2) I (cid:17) I ⊂ V . In particular, s , s (1) ∅ ,and s (2) ∅ are three distinct variables.Let η , η , and η (cid:16) η be the incidence representations of H , H , and H (cid:16) H ,respectively. From (5.7) (with D = ∅ ), we obtain ζ ask ( η (cid:16) η ) O ( s ) = 11 − t Z (cid:16) ( V ,V ) (cid:16) s − n + m − , (cid:16) − µ (1) I (cid:17) I ⊂ V , (cid:16) − µ (2) I (cid:17) I ⊂ V (cid:17) . Recursive formulae. This identity allows us to relate the rational functions W H ( X, T ), W H ( X, T ), and W H ( X, T ). We first express Z (cid:16) ( n ,n ) ( s ) in terms of (translates of) thefunctions Z V i ( s , s ( i ) ); cf. (5.5). Let t := q − s . Proposition 5.16. Z (cid:16) ( V ,V ) ( s ) = (cid:16) q − n − t − Z V ( s + n , s (1) )(1 − q − n − t )+ Z V ( s + n , s (2) )(1 − q − n − t ) (cid:17) / (1 − q − t ) (5.20)47 Ask zeta functions of hypergraphs Proof. It suffices to analyse the function I (cid:16) ( V ,V ) ( s ) := Z ( O V ) × × P | y | s Y I ⊂ V k x (1) I , x (2) , y k s I Y I ⊂ V k x (2) I , x (1) , y k s I d µ O V × O ( x, y ) , where x ( i ) I i = ( x ( i ) j : j ∈ I i ) and x = ( x (1) , x (2) ) = ( x (1)1 , . . . , x (1) n , x (2)1 , . . . , x (2) n ). Indeed, Z (cid:16) ( V ,V ) ( s ) = 1 + (1) − I (cid:16) ( V ,V ) ( s ); see (5.5). We proceed by decomposing the domain ofintegration of this function. On the set S × P for S := n ( x (1) , x (2) ) ∈ ( O V ) × : x (1) x (2) (mod P ) o , the integral is very simple. Indeed, µ (( O V ) × \ S ) = ( n ) q − n + ( n ) q − n whence(1) − Z S × P | y | s Y I ⊂ V k x (1) I , x (2) , y k s I Y I ⊂ V k x (2) I , x (1) , y k s I d µ O V × O ( x, y ) = (cid:0) − q − n − ( n ) q − n − ( n ) q − n (cid:1) gp (cid:16) q − t (cid:17) by (5.4). It remains to deal with Z ( O V ) × \ S × P | y | s Y I ⊂ V k x (1) I , x (2) , y k s I Y I ⊂ V k x (2) I , x (1) , y k s I d µ O V × O ( x, y )= Z P V × ( O V ) × × P | y | s Y I ⊂ V k x (2) I , x (1) , y k s I d µ O V × O ( x, y )+ Z ( O V ) × × P V × P | y | s Y I ⊂ V k x (1) I , x (2) , y k s I d µ O V × O ( x, y )= I V ,V ( s , s (2) ) + I V ,V ( s , s (1) );cf. (5.1). For i = 1, by invoking (5.5) again and also Theorem 5.5, we obtain I V ,n ( s , s (1) ) = (1) Z V ,V ( s , s (1) ) − − q − n − t − q − t ! = (1) 1 − q − n − t − q − t (cid:16) Z V ( s + n , s (1) ) − (cid:17) ;the argument for i = 2 is analogous. (cid:7) We now obtain the following expression for the rational function W H (cid:16) H ( X, T ) associ-ated with the complete union H (cid:16) H of the hypergraphs H and H .48 Ask zeta functions of hypergraphs Corollary 5.17. W H (cid:16) H ( X, T ) = ( X − m T − W H ( X, X − m T )(1 − X − m T )(1 − X n − m T )+ W H ( X, X − m T )(1 − X − m T )(1 − X n − m T )) / ((1 − T )(1 − X n − m T )) . (5.21) In particular, if H is the block hypergraph BH n ,m , then W H (cid:16) H ( X, T ) = W H ( X, X − m T ) (1 − X − m T )(1 − X n − m T )(1 − T )(1 − X n − m T ) . (5.22) Proof. Write t = q − s . As ζ ask ( η (cid:16) η ) O ( s ) = W H (cid:16) H ( q, t ) = 11 − t Z (cid:16) ( V ,V ) (cid:16) s − n + m − , ( − µ (1) I ) I ⊂ V , ( − µ (2) I ) I ⊂ V (cid:17) , we seek to describe the effect of replacing s by s − n + m − s ( i ) I i by − µ ( i ) I i ineach of the two functions Z V i ( s + n − i , s ( i ) ) in (5.20). Since ζ ask η O i ( s ) = W H i ( q, t ) = 11 − t Z V i (cid:16) s − n i + m i − , (cid:16) − µ ( i ) I i (cid:17) I i ⊂ V i (cid:17) by (5.7), we obtain Z V i (cid:16) s − n + m − n − i , (cid:16) − µ ( i ) I i (cid:17) I i ⊂ V i (cid:17) = W H i ( q, q − m − i t ) (1 − q − m − i t ) . This establishes the first claim. The special case follows from a simple computation using W BH n ,m ( X, T ) = (1 − X − m T ) / ((1 − T )(1 − X n − m T )); see Example 5.10(i). (cid:7) Given hypergraphs H , . . . , H r , repeated application of Corollary 5.17 yields explicitformulae for W H ⊕···⊕ H r ( X, T ) in terms of W H ( X, T ) , . . . , W H r ( X, T ). Let n , m ∈ N r . The main result of this section, namely Corollary 5.19, provides anexplicit formula for the rational function W H ( X, T ) associated with H := PH n , m = PH n ,m (cid:16) · · · (cid:16) PH n r ,m r , the reflection of BH n , m ; see (3.3)–(3.4). Note that n × m − diag ( n × m , . . . , n r × m r ) ∈ M n × m ( Z )is an incidence matrix of H ; up to reordering rows and columns, this is the general formof incidence matrices of hypergraphs with codisjoint supports (i.e. whenever µ I µ J = 0for I, J ⊂ V , then I = J or I c ∩ J c = ∅ ) and which also satisfy T µ I > I = ∅ .Let V i be the set of vertices of PH n i ,m i and let V = V t · · · t V r be that of H = PH n , m .49 Ask zeta functions of hypergraphs An auxiliary function. Consider the specialisation Z (cid:16) , codis V ( s ) := Z (cid:16) , codis V (cid:16) s ; s V c , . . . , s V c r (cid:17) := Z V (cid:16) s , (cid:16) δ ∃ i ∈ [ r ]: I = V c i s I (cid:17) I ⊂ V (cid:17) of Z V ( s ). In other words, Z (cid:16) , codis V ( s ) is obtained from Z V ( s ) by setting all variables s I to zero, except for those with I = V c i for some i .Let n = n + · · · + n r and m = m + · · · + m r . Let η (cid:16) n , m be the incidence representationof H . From (5.7) (with D = ∅ ), we obtain the identity ζ ask ( η (cid:16) n , m ) O ( s ) = 11 − t Z (cid:16) , codis V (cid:16) s − n + m − − m , . . . , − m r (cid:17) . (5.23)As before, we write t I := q − s I . An explicit formula and its consequences Theorem 5.18. Z (cid:16) , codis V ( s ) = 11 − q − t − q − n − t − r X i =1 ( n i ) q n i (cid:16) t V c i − (cid:17) − q − n − n i t t V c i . Prior to proving Theorem 5.18, we record our main result here, namely the followingimmediate consequence of Theorem 5.18 and (5.23). Corollary 5.19. W PH n , m ( X, T ) = 1(1 − T )(1 − X n − m T ) − X − m T − r X i =1 ( X n i − X m i − − X n i + m i − m T !! . (cid:7) Note that the formulae in Corollaries 5.17 and 5.19 indeed coincide where they overlap. Remark 5.20. Using “Big Theta Notation”, the estimate from [3] for the n th Fubininumber f n cited in (1.4) implies that f n = Θ (cid:16) n !(log 2) n +1 (cid:17) . We have thus produced explicitformulae of three (generally strictly decreasing) complexities:(i) For a general hypergraph H on n vertices, Corollary 5.6 expresses W H ( X, T ) as asum of | d WO n | = Θ (cid:16) n !(log 2) n +1 (cid:17) rational functions.(ii) If H = BH n , m for n , m ∈ N r , then Corollary 5.14 expresses W H ( X, T ) as a sum ofΘ (cid:16) r !(log 2) r +1 (cid:17) rational functions.(iii) Finally, if H = PH n , m is the reflection (see §3.1) of BH n , m , then Corollary 5.19expresses W H ( X, T ) as a sum of Θ( r ) rational functions.50 Ask zeta functions of hypergraphs Writing m = | E( H ) | , the rational functions appearing in these sums are products of O ( n )factors of the form ± X a , ± X a T , 1 − X a , and (1 − X a T ) ± for a ∈ Z with | a | = O ( n + m ).For another point view, write each of the above formulae over a common denominator.Then we saw that the denominators can (essentially) be written as products of O (2 n )factors of the form 1 − X A T in case (i), products of O (2 r ) such factors in case (ii), and asproducts of O ( r ) factors in case (iii). While cancellations may reduce the actual numberof factors for any given hypergraph, experiments suggest that our bounds generallyindicate the correct order of magnitude. Proof of Theorem 5.18. The following observation will be helpful. Lemma 5.21. For all N ∈ N , Z P N × P | y | a k x , . . . , x n − , y k a N − d µ O N × O ( x, y ) = q − a − a N − − ( N +1) (1) (1 − q − a − N )(1 − q − a − )(1 − q − a − a N − − N ) . Proof. This is a straightforward corollary of [57, Lemma 5.8] which implies that F N ( a , , . . . , , a N − , 0) := Z O N × O | y | a k x , . . . , x N − , y k a N − d µ O N × O ( x, y ) =(1)(1 − q − a − N )(1 − q − a − )(1 − q − a − a N − − N ) . (cid:7) Proof of Theorem 5.18. Consider Z (cid:16) , codis V ( s ; s V c , . . . , s V c r ) = 1 + (1) − Z ( O V ) × × P | y | s r Y i =1 k x V c i ; y k s V c i d µ O V × O ( x, y ) . Note that the product in the integrand is trivial unless x ∈ ( O V ) × has its P -adic unitsconcentrated in exactly one of the sets V i . We therefore split up the first factor of thedomain of integration in the form ( O V ) × = S t (( O V ) × \ S ), where S := n x ∈ ( O V ) × : { j ∈ [ r ] : ∃ v ∈ V j : x v ∈ O × } > o . Clearly µ (( O V ) × \ S ) = P ri =1 ( n i ) q n i − n whence(1) − Z S × P | y | s r Y i =1 k x V c i ; y k s V c i d µ O V × O ( x, y ) = (cid:16) − q − n − r X i =1 ( n i ) q n i − n (cid:17) gp (cid:16) q − t (cid:17) . (5.24)51 Ask zeta functions of hypergraphs By applying Lemma 5.21 for each j ∈ [ r ] (with N = n − n i + 1, a = s , a N − = s V c i ),we obtain (1) − Z ( O V ) × \ S × P | y | s r Y i =1 k x V c i ; y k s V c i d µ O V × O ( x, y )= (1) − r X i =1 ( n i ) q Z P n − ni +1 × P | y | s k x [ n − n i ] ; y k s V c i d µ O n − ni +1 × O ( x, y )= r X i =1 ( n i ) (1 − q − n − n i t )(1 − q − t ) gp (cid:16) q − n − n i t t V c i (cid:17) . (5.25)Combining (5.24) and (5.25) yields, after some trivial simplifications, that indeed Z (cid:16) , codis V ( s ; s V c , . . . , s V c r ) = 1 + (cid:16) − q − n − r X i =1 ( n i ) q n i − n (cid:17) gp (cid:16) q − t (cid:17) + r X i =1 ( n i ) (1 − q − n − n i t )(1 − q − t ) gp (cid:16) q − n − n i t t V c i (cid:17) = 11 − q − t − q − n − t − r X i =1 ( n i ) q n i (cid:16) t V c i − (cid:17) − q − n − n i t t V c i . (cid:7) In this section, we study four fundamental operations for hypergraphs: insert either a rowor a column of either all 0s or all 1s into any incidence matrix. In Proposition 5.23, werecord the effects of these operations on associated ask zeta functions. For group-theoreticapplications of these results, see §8.Throughout, let µ = ( µ I ) I ⊂ V be the vector of hyperedge multiplicities of a hypergraph H on the vertex set V ; see Definition 3.1 and the comments that follow it. Let n = | V | and m = | E( H ) | = P I ⊂ V µ I . Let • be a singleton set disjoint from V . Definition 5.22. We define(i) µ = ( ν J ) J ⊂ V t• by ν J = ( µ I , if J = I t • for I ⊂ V, , otherwise,(ii) µ = ( ν J ) J ⊂ V t• by ν J = ( µ J , if J ⊂ V, , otherwise,(iii) µ = (cid:16) µ I + δ I = V (cid:17) I ⊂ V , and(iv) µ = (cid:16) µ I + δ I = ∅ (cid:17) I ⊂ V . 52 Ask zeta functions of hypergraphs In other words, beginning with an arbitrary incidence matrix of H , we obtain associatedhypergraphs H := H ( µ ) by inserting a -row, H := H ( µ ) by inserting a -row, H := H ( µ ) by inserting a -column, H := H ( µ ) by inserting a -column.Note that H ( µ ) = H ( µ , • ) in the sense of Definition 5.1. We write µ (0) = µ and, for r ∈ N , µ ( r ) = ( µ ( r − ) . Likewise, we write H (0) = H and H ( r ) = (cid:16) H ( ( r − ) (cid:17) . We useanalogous notation for the other three operations. All four operations turn out to havetame effects on the ask zeta functions associated with H . Proposition 5.23. W H ( X, T ) = 1 − X n − m T − X n − m T W H ( X, T ) , (5.26) W H ( X, T ) = W H ( X, XT ) , (5.27) W H ( X, T ) = 1 − X − T − T W H ( X, X − T ) , (5.28) W H ( X, T ) = W H ( X, T ) . (5.29) Proof. The statement about W H ( X, T ) and W H ( X, T ) follow from [57, §3.4], the othersby inspection of (5.11) (with d = 1 for W H ( X, T )). (cid:7) Remark 5.24. For the purpose of determining W H ( X, T ) for a hypergraph H ≈ H ( µ ),Proposition 5.23 allows us to assume that µ satisfies µ V = µ ∅ = 0, T µ I > I = ∅ , and S µ I > I = V . In other words, we may assume that no incidence matrix of H ( µ ) has rowsor columns comprised exclusively of 0s or 1s. Conversely, by adding suitable rows orcolumns of s, Proposition 5.23 also allows us to e.g. assume that incidence matrices ofhypergraphs are squares; cf. [57, Cor. 3.7].We may now continue the story that began in Example 1.6. Example 5.25 (Example 1.6, part II) . Let H be the hypergraph on 8 vertices withincidence matrix (1.8) in Example 1.6. We are now in a position to compute therational function W H ( X, T ). Indeed, H is isomorphic to ( BH , ⊕ BH , ) (cid:16) BH , . Usingequations (5.19) (with n = n = 3 and m = m = 2) and (5.29), we obtain W ( BH , ⊕ BH , ) ( X, T ) = 1 + X − T − X − T − X − T + XT + X − T (1 − T )(1 − XT )(1 − X T ) . Therefore, by (5.22), W H ( X, T ) = W ( BH , ⊕ BH , ) (cid:16) BH , ( X, T )= W ( H (3) ⊕ H (3) ) ( X, X − T ) (1 − X − T )(1 − X − T )(1 − T )(1 − XT )= 1 + X − T − X − T − X − T + X − T + X − T (1 − T ) (1 − XT ) . (5.30)53 Ask zeta functions of hypergraphs Note that W H ( X, T ) coincides with the formula for W − Γ ( X, T ) given in (1.6). We will beable to explain this following our proof of the Cograph Modelling Theorem (Theorem D)in §7; see Example 7.28. Let K be a number field with ring of integers O = O K . Let ζ K ( s ) be the Dedekind zetafunction of K . As in §1.3, let V K be the set of non-Archimedean places of K and, for v ∈ V , let O v be the valuation ring of the v -adic completion of K . Let q v be the residuefield size of O v .Let η be the incidence representation (see §3.2) of a hypergraph H = H ( µ ) on a set V of cardinality n (cid:62) 1, where µ = ( µ I ) I ⊂ V ∈ N P ( V )0 is a vector of hyperedge multiplicities.By [57, Proposition 3.4], ζ ask η O ( s ) = Y v ∈V K ζ ask η O v ( s ) = Y v ∈V K W H ( q v , q − sv ) . The explicit formula for W H ( X, T ) in Corollary 5.6 allows us to deduce the following. Theorem 5.26. Let m := P ∅ = I ⊂ V µ I be the number of non-empty hyperedges of H .(i) For each compact DVR O , the real parts of the poles of ζ ask η O ( s ) are contained in P H := n | J | − X I ∩ J = ∅ µ I : J ⊂ V o ⊂ { − m , − m , . . . , n − , n } , a set of integers (!) of cardinality at most min { n , n + m } .(ii) The abscissa of convergence α ( H ) of ζ ask η O K ( s ) is a positive integer. It satisfies α ( H ) (cid:54) n + 1 and is independent of K .Proof. First note that if m = 0, then ζ ask η O ( s ) = 1 / (1 − q n − s ) and ζ ask η O K ( s ) = ζ K ( s − n );cf. [57, p. 577]. As n (cid:62) 1, both claims then follow immediately. Henceforth, supposethat m > | J | − P I ∩ J = ∅ µ I ∈ { − m , . . . , n } for each J ⊂ V . Since ζ ask η O ( s ) = W H ( q, q − s ), part (i) thus follows from Corollary 5.6 (with D = ∅ ).For part (ii), we first paraphrase (5.12) in the form W H ( X, T ) = 11 − T N X i =1 f i ( X − ) Y j ∈ I i gp (cid:16) X A ij T (cid:17) (5.31)for some N ∈ N , non-empty subsets I i ⊂ N , non-constant polynomials f i ( Y ) ∈ Z [ Y ]with constant term f i (0) = 1, and A ij ∈ P H . We may assume that N > W H ( X, T ) = 1 + ∞ P k =1 ∞ P l = −∞ a lk X l ! T k (1 − T ) Q i,j (1 − X A ij T ) . Ask zeta functions of hypergraphs As a product of finitely many translates of ζ K ( s ), the Euler product Y v ∈V K − T ) Q i,j (1 − X A ij T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = q v ,T = q − sv = ζ K ( s ) Y i,j ζ K ( s − A ij )has abscissa of convergence α := max { , A ij + 1 : i ∈ [ N ] , j ∈ I i } (cid:54) n + 1, where theestimate follows as in (i). Moreover, this product may be analytically continued to ameromorphic function on the whole complex plane. It thus suffices to show that theabscissa of convergence, α say, of the Euler product N H ( s ) := Y v ∈V K ∞ X k =1 ∞ X l = −∞ a lk X l T k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = q v ,T = q − sv (5.32)is strictly less than α . By [22, Lemma 5.4], α (cid:54) max (cid:26) l + 1 k : l ∈ Z , k ∈ N , a lk = 0 (cid:27) = max (cid:8) α , α (cid:62) (cid:9) , where α = sup { l + 1 : l ∈ Z , a l = 0 } and α (cid:62) = sup (cid:26) l + 1 k : l ∈ Z , k ∈ N (cid:62) , a lk = 0 (cid:27) . The coefficient of T in each Euler factor on the right-hand side of (5.32) is ∞ X l = −∞ a l X l = X i,j (cid:16) f i ( X − ) − (cid:17) X A ij . Hence, by the aforementioned properties of the polynomials f i ( Y ), we conclude that α < α . Next, for each subset S ⊂ I × · · · × I N with | S | (cid:62) P ( i,j ) ∈ S A ij | S | < P ( i,j ) ∈ S (1 + A ij ) | S | (cid:54) max { A ij : ( i, j ) ∈ S } (cid:54) α whence α (cid:62) < α . The independence of α ( H ) from K has been established, in greatergenerality, in [57, Theorem 4.20]. (cid:7) Remark 5.27. (i) As we mentioned in Remark 5.20, for specific µ the formula (5.12) may simplifydue to cancellations, possibly leading to a much smaller set of real parts of polesthan P H . Based on experimental evidence, however, for suitably “generic” µ , weexpect most of these at most 2 n candidate real poles in P H to survive cancellation.(ii) Every integer from 1 up to (and including) n +1 arises as the abscissa of convergenceof the ask zeta function of a hypergraph on n vertices. Indeed, α ( BH n,m ) =max { , n − m + 1 } ; see Example 5.10((i)) and cf. [57, Example 3.5].55 Uniformity for ask zeta functions of graphs (iii) The fact that both α ( H ) and all elements of P H are integers seems noteworthy.Indeed, the abscissae of convergence of Dirichlet generating functions arising fromrelated counting problems in subgroup or representation growth tend to be rational but typically non-integral numbers; cf. [25, Theorem 1.3 and §6] and [56, Theo-rem A(ii)] for (non-)integrality results in the area of subgroup and submodule zetafunctions and, for instance, [1, Theorem 1.2], [26, Corollary B], and [64, Theo-rem 4.22] in the context of representation zeta functions.(iv) Example 1.6 shows that, in general, integrality statements such as those in The-orem 5.26 hold neither for the (Euler products of instances of the) functions W +Γ ( X, T ) featuring in Theorem A(iii) nor for the functions W − Γ ( X, T ) in Theo-rem A(ii), unless Γ is a cograph; cf. Theorem D and see Question 1.8.For staircase hypergraphs (§5.1.1), we can considerably strengthen Theorem 5.26(ii).Indeed, inspection of (5.14) yields the following result. Proposition 5.28. Let m = ( m , . . . , m n ) ∈ N n +10 . Let ση m denote the incidencerepresentation of the staircase hypergraph ΣH m ; see (3.9) . Then for each number field K with ring of integers O K , the abscissa of convergence of ζ ask ση m ( s ) is given by α ( ΣH m ) = max n , n − j − X ι>j m ι : j = 0 , . . . , n − o . Moreover, the function ζ ask ση O K m ( s ) may be meromorphically continued to the whole of C . (cid:7) The main result of this section, Theorem 6.4, establishes that, subject to very mild as-sumptions, a simultaneous generalisation of the two types of adjacency modules from §3.3is torically combinatorial (see §4.4). This will, in particular, provide a constructive proofof Theorem A(ii)–(iii).In §6.1, we develop the general setup for the class of adjacency modules that appear inTheorem 6.4. In §6.2, we describe several graph-theoretic operations with tame effectson adjacency modules. In §6.3, we show that “torically torically combinatorial” and“torically combinatorial” are equivalent properties. Both §§6.2–6.3 are then employed inthe proof of Theorem 6.4 in §6.4.Throughout, let R be a ring. Definition 6.1. A weighted signed multigraph ( WSM ) (over σ ) is a quadruple Γ = (Γ , σ, wt , sgn) , where 56 Uniformity for ask zeta functions of graphs ( W1 ) Γ = ( V, E, | · | ) is a multigraph (see §3.1),( W2 ) σ ⊂ R (cid:62) V is a cone,( W3 ) wt is a function E → Z V with u + wt( e ) ∈ σ ∗ for all e ∈ E and u ∈ | e | , and( W4 ) sgn is a function E → {± } .Henceforth, let Γ be a WSM as above. Let X = ( X v ) v ∈ V as before. For u, v ∈ V and ω ∈ Z V , define [ u, v ; ± , ω ] R := ( X u + ω v ± X v + ω u, if u = v, ± X u + ω u, if u = v, an element of R [ X ± ] V ; we usually drop the subscript R in the following. Note thatif u = v , then [ v, u ; ± , ω ] = ± [ u, v ; ± , ω ]. Further note that [ u, v ; ± , 0] = [ u, v ; ± W3 ) in Definition 6.1, for e ∈ E and u ∈ | e | , we have X u +wt( e ) ∈ R σ ; see §4.2 fora definition of R σ . Let adj ( Γ ; R ) := D [ u, v ; sgn( e ) , wt( e )] R : e ∈ E with | e | = { u, v } E (cid:54) R σ V. The adjacency module of Γ over R is the R σ -moduleAdj( Γ ; R ) := R σ V adj ( Γ ; R ) . Remark 6.2. (i) If Γ is a graph, then Adj(Γ , R (cid:62) V, , ± R ) coincides with the adjacency moduleAdj(Γ , ± R ) of Γ over R as defined in §3.3. (Here, we assume that Γ is simple inthe negative case.)(ii) Of course, the signs of loops have no effect on adjacency modules. They are includedfor notational convenience only. Lemma 6.3. (i) Adj( Γ ; S ) = Adj( Γ ; R ) S σ for each ring map R → S .(ii) Let Γ be the WSM obtained from Γ by replacing σ by a cone τ ⊂ σ . Then Adj( Γ ; R ) ≈ R τ Adj( Γ ; R ) ⊗ R σ R τ . (cid:7) The following result and its constructive proof constitute the main contribution of thepresent section; a proof will be given in §6.4. Theorem 6.4. Let Γ = (Γ , σ, wt , sgn) be a weighted signed multigraph. Let R be a ring.Suppose that one of the following conditions is satisfied: (i) ∈ R × . (ii) in R .(iii) sgn ≡ − (irrespective of R ). Then Adj( Γ ; R ) is torically combinatorial (see §4.4). Theorem 6.4 easily implies Theorem A: 57 Uniformity for ask zeta functions of graphs Proof of Theorem A. Part (i) was already proved in §4.4. For (ii)–(iii), combine Proposi-tion 3.7, Proposition 4.8, Remark 6.2(i), and Theorem 6.4 (with R = Z or R = Z [1 / (cid:7) Proof of Corollary B. Combine Theorem A and Proposition 3.9. (cid:7) While the preceding two proofs only applied Theorem 6.4 in the special case that Γ arises as in Remark 6.2, our recursive proof of Theorem 6.4 heavily relies on the greatergenerality developed here.A particularly easy special case of Theorem 6.4 deserves to be spelled out at this point.We say that a graph is solitary if each of its edges is a loop. Proposition 6.5. Let σ ⊂ R (cid:62) V be a cone. Let Γ be a WSM over σ (with underlyingvertex set V ) such that the underlying graph of Γ is solitary. Then Adj( Γ ; R ) is acombinatorial R σ -module. (cid:7) Informally, our proof of Theorem 6.4 given in §6.4 proceeds by induction on an invariantwhich measures to what extent a graph fails to be solitary; the base case of our inductionwill be provided by Proposition 6.5. As we will see in this subsection, subject to various assumptions, we may modify theedges (as well as their weights and signs) of weighted signed multigraphs without affectingthe isomorphism type of the associated adjacency module. This will constitute the heartof our proof of Theorem 6.4 in §6.4.Throughout let Γ = (Γ , σ, wt , sgn) be a WSM , where Γ = ( V, E, | · | ). An incidentpair of Γ is a pair (“formal product”) u.e , where e ∈ E and u ∈ | e | . Recall that (cid:54) σ denotes the preorder on Z V such that u (cid:54) σ v if and only if v − u ∈ σ ∗ . Given incidentpairs u.e and w.f of Γ, we say that u.e dominates w.f (in Γ ) if u + wt( e ) (cid:54) σ w + wt( f ).The next three lemmas and their proofs (nearly) follow an identical pattern: subject todominance conditions, suitable edges of Γ can be transplanted to produce a new WSM Γ such that adj ( Γ ; R ) = adj ( Γ ; R ) for every ring R . The effects of the operations Γ (cid:32) Γ onthe underlying multigraphs are indicated in Figure 1. Note that these figures only depictthose parts of the respective multigraphs that are relevant for the result in question. Lemma 6.6 (Dominant loop vs non-loop) . Let u, v ∈ V be distinct. Let ‘, h ∈ E with | ‘ | = { u } and | h | = { u, v } . Suppose that u.‘ dominates v.h . Define a multigraph Γ := ( V, E, k · k ) , where k e k := ( | e | , if e = h, { v } , if e = h. Define wt : E → Z V via wt ( e ) := ( wt( e ) , if e = h,u − v + wt( h ) , if e = h. Then the following hold: Uniformity for ask zeta functions of graphs (i) Γ := (Γ , σ, wt , sgn) is a WSM .(ii) adj ( Γ ; R ) = adj ( Γ ; R ) for every ring R ; in particular, Adj( Γ ; R ) = Adj( Γ ; R ) .Proof. (i) We need to check that x + wt ( e ) ∈ σ ∗ for all e ∈ E and x ∈ k e k . For e = h , thisclearly follows since Γ is a WSM . It also follows in the remaining case e = h since v + wt ( h ) = u + wt( h ) ∈ σ ∗ , again since Γ is a WSM .(ii) Let I := D [ x, y ; sgn( e ) , wt( e )] : e ∈ E \ { h } with | e | = { x, y } E ⊂ adj ( Γ ; R ) ∩ adj ( Γ ; R ) . Further define a := [ u, v ; sgn( h ) , wt( h )] ∈ adj ( Γ ; R ) ,a := [ v, v ; sgn( h ) , wt ( h )] ∈ adj ( Γ ; R ) , and b := [ u, u ; sgn( ‘ ) , wt( ‘ )] ∈ I and note that adj ( Γ ; R ) = h a i + I and adj ( Γ ; R ) = h a i + I . Since u.‘ dominates v.h ,we have t := X v +wt( h ) − u − wt( ‘ ) ∈ R σ . Therefore,sgn( h ) a = X u +wt( h ) v = a − sgn( h ) sgn( ‘ ) tb, whence a ≡ ± a (mod I ). (cid:7) Lemma 6.7 (Dominant non-loop vs loop) . Let u, v ∈ V be distinct. Let ‘, h ∈ E with | ‘ | = { v } and | h | = { u, v } . Suppose that u.h dominates v.‘ . Define a multigraph Γ := ( V, E, k · k ) , where k e k := ( | e | , if e = ‘, { u } , if e = ‘. Define wt : E → Z V via wt ( e ) := ( wt( e ) , if e = ‘, v − u + wt( ‘ ) , if e = ‘ and sgn : E → {± } via sgn ( e ) := ( sgn( e ) , if e = ‘, − sgn( h ) sgn( ‘ ) , if e = ‘. Then the following hold:(i) Γ := (Γ , σ, wt , sgn ) is a WSM . v‘ h Γ (cid:32) u v‘ h Γ (a) Lemma 6.6: dominant loop vs non-loop u vh ‘ Γ (cid:32) u v‘ h Γ (b) Lemma 6.7: dominant non-loop vs loop u v wh i Γ (cid:32) u vwhi Γ (c) Lemma 6.8: dominant non-loop vs non-loop u vhi Γ (cid:32) u vh Γ (d) Lemma 6.9: parallel edges I u vhi Γ (cid:32) u vh i Γ (e) Lemma 6.10: parallel edges II u vh Γ (cid:32) u v h Γ (f) Lemma 6.11: trimming spikes Figure 1: Illustrations of Lemma 6.6–6.11 Uniformity for ask zeta functions of graphs (ii) adj ( Γ ; R ) = adj ( Γ ; R ) for every ring R ; in particular, Adj( Γ ; R ) = Adj( Γ ; R ) .Proof. (i) We need to check that x + wt ( e ) ∈ σ ∗ for all e ∈ E and x ∈ k e k . For e = ‘ and x ∈ k e k = | e | , we have x + wt ( e ) = x + wt( e ) ∈ σ ∗ since Γ is a WSM . Moreover, v + wt( h ) ∈ σ ∗ since Γ is a WSM and v + wt( ‘ ) − u − wt( h ) ∈ σ ∗ since u.h dominates v.‘ . Hence, u + wt ( ‘ ) = 2 v − u + wt( ‘ ) ∈ σ ∗ .(ii) Let I := D [ x, y ; sgn( e ) , wt( e )] : e ∈ E \ { ‘ } with | e | = { x, y } E ⊂ adj ( Γ ; R ) ∩ adj ( Γ ; R ) . Further define a := [ u, v ; sgn( h ) , wt( h )] ∈ I,b := [ v, v ; sgn( ‘ ) , wt( ‘ )] ∈ adj ( Γ ; R ) , and b := [ u, u ; sgn ( ‘ ) , wt ( ‘ )] ∈ adj ( Γ ; R )and note that adj ( Γ ; R ) = h b i + I and adj ( Γ ; R ) = h b i + I . Since u.h dominates v.‘ ,we have t := X v +wt( ‘ ) − u − wt( h ) ∈ R σ . Therefore, b − sgn( ‘ ) ta = b , whence b ≡ b (mod I ). (cid:7) Lemma 6.8 (Dominant non-loop vs non-loop) . Let u, v, w ∈ V be distinct. Let h ∈ E with | h | = { u, v } and i ∈ E with | i | = { v, w } . Suppose that u.h dominates w.i . Define amultigraph Γ := ( V, E, k · k ) , where k e k := ( | e | , if e = i, { u, w } , if e = i. Define wt : E → Z V via wt ( e ) := ( wt( e ) , if e = i,v − u + wt( i ) , if e = i and sgn : E → {± } via sgn ( e ) := ( sgn( e ) , if e = i, − sgn( h ) sgn( i ) , if e = i. Then the following hold:(i) Γ := (Γ , σ, wt , sgn ) is a WSM .(ii) adj ( Γ ; R ) = adj ( Γ ; R ) for every ring R ; in particular, Adj( Γ ; R ) = Adj( Γ ; R ) . Uniformity for ask zeta functions of graphs Proof. (i) We need to check that x + wt ( e ) ∈ σ ∗ for all e ∈ E and x ∈ k e k . For e = i and x ∈ k e k = | e | , x + wt ( e ) = x + wt( e ) ∈ σ ∗ . As Γ is a WSM , each of v + wt( h ), v + wt( i ), and w + wt( i ) belongs to σ ∗ . Since u.h dominates w.i , we conclude that u + wt ( i ) = v + wt( i ) ∈ σ ∗ and w + wt ( i ) = w + v − u + wt( i ) = ( v + wt( h )) + ( w + wt( i )) − ( u + wt( h )) ∈ σ ∗ . (ii) Let I := D [ x, y ; sgn( e ) , wt( e )] : e ∈ E \ { i } with | e | = { x, y } E ⊂ adj ( Γ ; R ) ∩ adj ( Γ ; R ) . Further define a := [ u, v ; sgn( h ) , wt( h )] ∈ I,b := [ v, w ; sgn( i ) , wt( i )] ∈ adj ( Γ ; R ) , and b := [ u, w ; sgn ( i ) , wt ( i )] ∈ adj ( Γ ; R )and note that adj ( Γ ; R ) = h b i + I and adj ( Γ ; R ) = h b i + I .Since u.h dominates w.i , we have t := X w +wt( i ) − u − wt( h ) ∈ R σ . Therefore, b − sgn( i ) ta = b , whence b ≡ b (mod I ). (cid:7) Lemma 6.9 (Parallel edges I) . Let h, i ∈ E be distinct with | h | = | i | . Suppose that wt( h ) (cid:54) σ wt( i ) . Define a multigraph Γ := ( V, E \ { i } , k · k ) and a WSM Γ := (Γ , σ, wt , sgn ) , where k · k , wt , and sgn are the restrictions of | · | , wt , and sgn to E \ { i } , respectively.Suppose that one of the following conditions is satisfied: (i) h (hence i ) is a loop.(ii) sgn( h ) = sgn( i ) . Then adj ( Γ ; R ) = adj ( Γ ; R ) for every ring R ; in particular, Adj( Γ ; R ) = Adj( Γ ; R ) .Proof. Write | h | = { u, v } . Since wt( h ) (cid:54) σ wt( i ), we have t := X wt( i ) − wt( h ) ∈ R σ . Theclaim follows since, in each one of the two cases listed, t · [ u, v ; sgn( h ) , wt( h )] = ± [ u, v ; sgn( i ) , wt( i )] . (cid:7) Lemma 6.10 (Parallel edges II) . Let u, v ∈ V be distinct. Let h, i ∈ E with | h | = | i | = { u, v } , sgn( h ) = − sgn( i ) , and wt( h ) (cid:54) σ wt( i ) . Define a multigraph Γ := ( V, E, k · k ) ,where k e k := ( | e | , if e = i, { v } , if e = i. Define wt : E → Z V via wt ( e ) := ( wt( e ) , if e = i,u − v + wt( i ) , if e = i Then the following hold: Uniformity for ask zeta functions of graphs (i) Γ := (Γ , σ, wt , sgn) is a WSM .(ii) adj ( Γ ; R ) = adj ( Γ ; R ) for every ring R in which is invertible; in particular, Adj( Γ ; R ) = Adj( Γ ; R ) for such rings R .Proof. (i) For e = i and x ∈ k e k = | e | , we have x + wt ( e ) = x + wt( e ) ∈ σ ∗ . As Γ is a WSM , u + wt( i ) belongs to σ ∗ whence v + wt ( i ) = u + wt( i ) ∈ σ ∗ .(ii) Let I := D [ x, y ; sgn( e ) , wt( e )] : e ∈ E \ { i } with | e | = { x, y } E ⊂ adj ( Γ ; R ) ∩ adj ( Γ ; R ) . Further define a := [ u, v ; sgn( h ) , wt( h )] ∈ I,b := [ u, v ; sgn( i ) , wt( i )] ∈ adj ( Γ ; R ) , and b := [ v, v ; sgn( i ) , wt ( i )] ∈ adj ( Γ ; R )and note that adj ( Γ ; R ) = h b i + I and adj ( Γ ; R ) = h b i + I .Since wt( h ) (cid:54) σ wt( i ), we have t := X wt( i ) − wt( h ) ∈ R σ . As b + ta = ± b , weconclude that adj ( Γ ; R ) = adj ( Γ ; R ) whenever 2 is invertible in R . (cid:7) By a spike of Γ , we mean a pair ( u, v ) of distinct vertices of Γ such that (i) u is theonly neighbour of v , (ii) there is only one edge e ∈ E with | e | = { u, v } , and (iii) u (cid:54) σ v . Lemma 6.11 (Trimming spikes) . Let ( u, v ) be a spike of Γ . Let h ∈ E be the uniqueedge with | h | = { u, v } . Define a multigraph Γ := ( V, E, k · k ) , where k e k := ( | e | , if e = h, { v } , if e = h. Define wt : E → Z V via wt ( e ) := ( wt( e ) , if e = h,u − v + wt( h ) , if e = h. (i) Γ := (Γ , σ, wt , sgn) is a WSM .(ii) Adj( Γ ; R ) ≈ R σ Adj( Γ ; R ) for every ring R . (However, in contrast to the precedinglemmas, adj ( Γ ; R ) and adj ( Γ ; R ) may differ.)Proof. (i) For e ∈ E \ { h } and x ∈ k e k = | e | , we have x + wt ( e ) = x + wt( e ) ∈ σ ∗ . Moreover, v + wt ( h ) = u + wt( h ) ∈ σ ∗ since Γ is a WSM .63 Uniformity for ask zeta functions of graphs (ii) Since u (cid:54) σ v , we obtain an R σ -module automorphism θ of R σ V given by xθ = ( x, if x = v,v − sgn( h ) X v − u u, if x = v. We now show that adj ( Γ ; R ) θ = adj ( Γ ; R ); the claim then follows immediately.Let e ∈ E \ { h } with | e | = { x, y } . Since ( u, v ) is a spike but e = h , we have v 6∈ | e | .Hence, θ fixes [ x, y ; sgn( e ) , wt( e )] (= [ x, y ; sgn( e ) , wt ( e )]). The claim follows since[ u, v ; sgn( h ) , wt( h )] θ = X u +wt( h ) v = ± [ v, v ; sgn( h ) , wt ( h )]. (cid:7) Corollary 6.12. Let the WSM Γ = (Γ , σ, wt , sgn ) be derived from Γ using any one ofLemmas 6.6–Lemma 6.9 or Lemma 6.11. If sgn ≡ − , then sgn ≡ − . (cid:7) Remark 6.13. Note that even if the underlying multigraph of a WSM admits no paralleledges, each of Lemmas 6.6–Lemma 6.8, Lemma 6.10, or Lemma 6.11 might introduceparallel edges. This section establishes the (intuitively evident) fact that a torically {torically combina-torial} module over a toric ring is itself torically combinatorial; see Corollary 6.16. Lemma 6.14. Let T be a non-empty finite set of cones in R V . Then there exists afan F in R V such that the following conditions are satisfied:(i) For each τ ∈ T , there exists Σ ⊂ F with τ = S Σ .(ii) For each σ ∈ F , there exists τ ∈ T with σ ⊂ τ .(iii) |F | = S T .Proof. For x ∈ R V , define x ± and x = as in the proof of Lemma 4.1. For each τ ∈ T , thereexists a non-empty finite set H τ ⊂ Z V such that τ = T h ∈ H τ h + . For h ∈ H := S τ ∈ T H τ ,define a complete fan F h := { h + , h − , h = } . Let F := V h ∈ H F h and F := { σ ∈ F : ∃ τ ∈ T. σ ⊂ τ } . Since F is a fan, so is F . We claim that F has the desired properties, (ii)being satisfied by construction.For (i), let τ ∈ T . Recall that τ = T h ∈ H τ h + . Let x ∈ τ be arbitrary. For each h ∈ H ,define σ x ( h ) ∈ F h via σ x ( h ) = ( h ± , if x ∈ h ± \ h = ,h = , if x ∈ h = ;in other words, σ x ( h ) is the unique cone in F h which contains x in its relative interior.Let σ x = T h ∈ H σ x ( h ) ∈ F and note that x ∈ σ x . Since x ∈ τ = T h ∈ H τ h + , for each h ∈ H τ ,we have σ x ( h ) ∈ { h + , h = } and hence σ x ( h ) ⊂ h + . Thus, σ x = \ h ∈ H σ x ( h ) ⊂ \ h ∈ H τ σ x ( h ) ⊂ \ h ∈ H τ h + = τ ;64 Uniformity for ask zeta functions of graphs in particular, σ x ∈ F . We may thus take Σ to be the finite (!) set { σ x : x ∈ τ } . Finally,by (i)–(ii), we have S T ⊂ |F | ⊂ S T . (cid:7) In particular, we can construct “fans of fans” as follows. Corollary 6.15. Let F be a fan of cones in R V . For each σ ∈ F , let F σ be a fan ofcones in R V with |F σ | = σ . Then there exists a fan F of cones in R V with the followingproperties:(i) F refines F .(ii) |F | = |F | .(iii) For each σ ∈ F and σ ∈ F σ , there exists Σ ⊂ F with σ = S Σ .(iv) For each σ ∈ F , there exist σ ∈ F and σ ∈ F σ with σ ⊂ σ .Proof. Let F be as in Lemma 6.14 with T := S σ ∈F F σ . Let F := F ∧ F . The firstproperty holds by definition and the second one since |F | = S T = |F | . For (iii), let σ ∈ F and σ ∈ F σ ⊂ T . By Lemma 6.14(i), there exists Σ ⊂ F with σ = S Σ. As σ ⊂ σ , we have σ = S Σ , where Σ := { % ∩ σ : % ∈ Σ } ⊂ F . For (iv), every cone in F is contained in a cone from F and each cone in F is contained in an element of T . (cid:7) Corollary 6.16. Let o ⊂ R (cid:62) V be a cone. Let M be an R o -module. Suppose that F isa fan in R (cid:62) V with |F | = o such that M ⊗ R o R σ is torically combinatorial over R σ foreach σ ∈ F . Then M is torically combinatorial as an R o -module.Proof. By assumption, for each σ ∈ F , there exists a fan F σ with support σ such that( M ⊗ R o R σ ) ⊗ R σ R τ ≈ R τ M ⊗ R o R τ is combinatorial over R τ for each τ ∈ F σ . Now applythe preceding corollary and note that a change of scalars of a combinatorial module alonga natural ring map R τ , → R τ (coming from an inclusion τ ⊂ τ of cones) preserves theproperty of being combinatorial. (cid:7) Let Γ = (Γ , σ, wt , sgn) be a WSM , where Γ = ( V, E, | · | ) and σ ⊂ R (cid:62) V .Define s (Γ) (the “social degree” of Γ) to be the number of non-loops of Γ in E . Notethat s (Γ) = 0 if and only if Γ is solitary (see §6.1). Let R be a ring. If 2 = 0 in R , thenAdj( Γ ; R ) does not depend on sgn( · ) at all so we may assume that sgn ≡ − Γ ; R ) is torically combinatorialwhenever sgn ≡ − R . We proceed by induction on s (Γ). Base case. If s (Γ) = 0, then Γ is solitary and Adj( Γ ; R ) is combinatorial (not merely torically combinatorial) by Proposition 6.5.Henceforth, suppose that s (Γ) > 0. 65 Uniformity for ask zeta functions of graphs General assumptions and reductions. We first carry out a number of general reductions;none of these increases s (Γ).The following operations amount to (i) constructing a fan F with support σ and(ii) considering the cases obtained by replacing σ by a cone in F ; this strategy is justifiedby Lemma 6.3 and Corollary 6.16.Thus, by shrinking σ via Lemma 4.1 and using Lemma 6.9, we may assume that Γ hasno parallel edges except possibly parallel non-loops with different signs. By shrinking σ yet further, we may also assume that for any two incident pairs of Γ, one of themdominates the other; see §6.2 for this notion. The last condition is clearly equivalent tothe preorder (cid:54) σ from §4.1 being total on elements x + wt( e ) ∈ Z V for e ∈ E and x ∈ | e | . Parallel non-loops with opposite signs. Suppose that Γ has parallel non-loops withopposite signs. In particular, sgn 6≡ − R . Let u, v ∈ V be distinct and let h, i ∈ E with | h | = | i | = { u, v } but sgn( h ) = sgn( i ). Our as-sumption on dominance of incidence pairs implies that wt( h ) (cid:54) σ wt( i ) or wt( h ) (cid:62) σ wt( i ).Without loss of generality, suppose that we are in the former case. Let Γ be the WSM obtained from Γ using Lemma 6.10. By construction, s (Γ ) < s (Γ). Indeed, the non-loop i of Γ is a loop of Γ and other edges coincide in the sense that they have the same supportin each multigraph. By induction, Adj( Γ ; R ) = Adj( Γ ; R ) is torically combinatorial.We may therefore assume that Γ has no parallel edges at all. Recall that we also assumethat given any two incident pairs of Γ, one of them dominates the other. Since s (Γ) > s (Ξ) > Using a dominant loop. Suppose that ‘ ∈ E is a loop at u ∈ V in Ξ such that u.‘ dominates each incident pair of Ξ. Since Ξ is not solitary but connected, it contains anon-loop h ∈ E with u ∈ | e | . Let Γ be the WSM obtained from Γ using Lemma 6.6.Since h is a loop of Γ and all other edges are unchanged as above, s (Γ ) < s (Γ). Hence,Adj( Γ ; R ) = Adj( Γ ; R ) is torically combinatorial by induction. Using a dominant non-loop. We may thus assume that u.h is an incident pair of Ξwhich dominates all incident pairs of Ξ and that h is not a loop. Write | h | = { u, v } andnote that u (cid:54) σ v since u.h dominates v.h .For each edge i ∈ E \ { h } with v ∈ | i | , we then obtain a WSM Γ as in Lemma 6.7 orLemma 6.8 with v Γ i and such that all other edges of Γ have the same support in Γand Γ . We may repeatedly apply these lemmas to all such edges i , one after the other,to derive a WSM ˜ Γ with Adj( Γ ; R ) = Adj( ˜ Γ ; R ) and such that the underlying graph ˜Γ of˜ Γ satisfies s (˜Γ) = s (Γ). By construction, ( u, v ) is then a spike of ˜ Γ . By deriving ˜ Γ from˜ Γ via Lemma 6.11, we obtain s (˜Γ ) < s (˜Γ) = s (Γ) whence Adj( ˜ Γ ; R ) ≈ R σ Adj( ˜ Γ ; R ) =Adj( Γ ; R ) is torically combinatorial by induction. Restrictions on R . We only made use of the assumption that 2 be invertible in R whenwe considered parallel edges with opposite signs. If all edge signs of a WSM Γ are − Graph operations and ask zeta functions of cographs then by Corollary 6.12, the same is true for all the graphs derived from Γ as part ofour inductive proof above. Hence, no restrictions on R are needed in this case and thiscompletes the proof of Theorem 6.4. (cid:7) Remark 6.17. Given a WSM Γ = (Γ , σ, wt , sgn) as above, our inductive proof ofTheorem 6.4 gives rise to a recursive algorithm for constructing a fan F with support σ andfor each τ ∈ F a WSM Γ τ with solitary underlying graph such that Adj( Γ ; R ) ⊗ R σ R τ ≈ R τ Adj( Γ τ ; R ) for each τ ∈ F (and subject to the assumptions on R from above). Togetherwith Proposition 4.8 and the techniques for computing monomial integrals from [54, 58],we thus obtain an algorithm for explicitly computing the rational functions in Theorem A.This algorithm turns out to be quite practical; see §9. Remark 6.18. In the setting of Theorem A(iii), the arguments developed in this sectiondo not apply to compact DVR s of characteristic 2 due to the factors ± DVR s O with residue characteristic 2. For example, usingeither the method from [57, §9.1] or the one developed here (see §9.1), we find that W +K ( X, T ) = ( T + T + 1 − X − T − X − T + 6 X − T + 3 X − T − X − T − X − T − X − T ) / (1 − T ) = 1 + (5 − X − + 3 X − − X − | {z } =: g ( X ) ) T + O ( T ) . On the other hand, a simple calculation shows that the average size of the kernel of amatrix of the form x yx zy z over F f is given by h (2 f ), where h ( X ) = 1 + X + X − ; note that g ( x ) = h ( x ) for all real x > 1. In particular, for each compact DVR O with residue field size q = 2 f , the function W +K ( q, q − s ) differs from the ask zeta function of the positive adjacency representationassociated with K over O . In this section, we deduce the Cograph Modelling Theorem (Theorem D) from a structuralresult (Theorem 7.1) which relates incidence modules of hypergraphs and adjacencymodules of cographs. After collecting some facts about cographs in §7.1, we formallystate Theorem 7.1 in §7.2. We give an outline of the latter theorem’s proof in §7.3 whichis then fleshed out in §§7.4–§7.7.Since we will focus exclusively on negative adjacency representations of simple graphs,in this section, we frequently omit references to the “negative” part. Throughout, R is aring, V is a finite set, and X = ( X v ) v ∈ V consists of algebraically independent variablesover R . All graphs are assumed to be simple in this section.67 Graph operations and ask zeta functions of cographs A cograph is a graph which belongs to the smallest class of graphs which containsisolated vertices and which is closed under both disjoint unions and joins of graphs. Inthis definition, “joins” can be replaced by “taking complements”. Cographs have appearedin various contexts and under various names such as “complement reducible graphs”and “ P -free graphs”; see [20]. They admit numerous equivalent characterisations; see[20, Theorem 2]. For instance, cographs are precisely those graphs all of whose connectedinduced subgraphs have diameter at most 2. Moreover, cographs are also precisely thosegraphs that do not contain a path on four vertices as an induced subgraph.As explained in [20], each cograph can be represented by a cotree : a rooted treewhose internal vertices are labelled using one of the symbols ⊕ and ∨ (corresponding todisjoint unions and joins, respectively) and whose leaves correspond to the vertices of thecograph. This representation is unique up to isomorphism of rooted trees provided that(i) each internal vertex has at least two descendants and (ii) adjacent internal verticesare labelled differently. Generalising the definition of incidence modules Inc( H ; R ) in §3.2, for a hypergraph H = ( V, E, | · | ) and cone σ ⊂ R (cid:62) V , we let inc ( H , σ ; R ) := D X v e : v ∼ H e ( v ∈ V, e ∈ E ) E (cid:54) R σ E and we define the incidence module of H with respect to σ over R to beInc( H , σ ; R ) := R σ E inc ( H , σ ; R ) . Clearly, Inc( H , σ ; R ) ≈ R σ Inc( H ; R ) ⊗ R R σ . (7.1)For a simple graph Γ with vertex set V and a cone σ ⊂ R (cid:62) V , we obtain a weightedsigned multigraph (see §6.1) Γ := (Γ , σ, , − adj (Γ , σ ; R ) := adj ( Γ ; R ) = D X v w − X w v : v, w ∈ V, v ∼ Γ w E (cid:54) R σ V and define the adjacency module of Γ with respect to σ over R to beAdj(Γ , σ ; R ) := Adj( Γ ; R ) = R σ V adj (Γ , σ ; R ) . To further simplify our notation, we let Adj(Γ; R ) := Adj(Γ , R (cid:62) V ; R ); note that thisnotation is consistent with §3.3. Observe thatAdj(Γ , σ ; R ) ≈ R σ Adj(Γ; R ) ⊗ R [ X ] R σ ; (7.2)cf. Lemma 6.3. In the following, we often omit R from our notation in case R = Z .The following is the main result of the present section.68 Graph operations and ask zeta functions of cographs Theorem 7.1 (Cograph Modelling Theorem: structural form) . Let Γ be a cograph.Let C be the set of connected components of Γ . Then there exists a hypergraph H with V(Γ) = V( H ) , | E( H ) | = | V(Γ) | − | C | , and such that Adj(Γ) and Inc( H ) ⊕ Z [ X ] C aretorically isomorphic Z [ X ] -modules, where X = ( X v ) v ∈ V . Our proof of this theorem in §7.6 below is based on a number of algebraic and graph-theoretic techniques developed in the following. Our proof is effective: given a cograph Γ,we can write down an explicit hypergraph H (a “model” of Γ in a sense to be formalisedin §7.6) as in Theorem 7.1. The final piece towards a proof of Theorem D is the followingcomparison result for adjacency and incidence representations. Lemma 7.2. Let Γ be a graph and let H be a hypergraph, both with common vertex set V .Let c (cid:62) and suppose that | E( H ) | = | V |− c . Let Σ be a set of cones with S Σ = R (cid:62) V andsuch that Adj(Γ , σ ) ≈ Z σ Inc( H , σ ) ⊕ Z cσ for each σ ∈ Σ . Then W − Γ ( X, T ) = W H ( X, T ) .Proof. Let γ (= γ − ) and η be the adjacency and incidence representation of Γ and H over Z , respectively; see §§3.2–3.3. As always, let O be a compact DVR . First, for each cone σ ⊂ R (cid:62) V and finitely generated O σ -module M , we clearly have ζ M ⊕ O σ ( s ) = ζ M ( s − R (cid:62) V . Now combine Proposition 3.3, Proposition 3.7, andLemma 4.6 to obtain ζ ask γ O ( s ) = (1 − q − ) − ζ Adj(Γ) ⊗ O [ X ] ( s ) = (1 − q − ) − ζ Inc( H ) ⊗ O [ X ] ( s − c ) = ζ ask η O ( s ) . (cid:7) We may now deduce the version of the Cograph Modelling Theorem from the intro-duction. Proof of Theorem D. Combine Theorem 7.1 and Lemma 7.2 with c = | C | . (cid:7) Remark 7.3. Assuming the validity of Theorem 7.1, we actually proved a slightlystronger result than Theorem D. Namely, for each cograph Γ, there exists a hypergraph H on the same set of vertices with | E( H ) | < | V(Γ) | and W − Γ ( X, T ) = W H ( X, T ). By repeatedapplication of (5.29), we may assume that | E( H ) | = | V(Γ) | − 1. The incidence matricesof H are then “near squares” in the sense that only one column is missing from a square.We note that the hypothesis of Theorem D itself is not optimal: Example 7.4. The rational function W − P ( X, T ) associated with a path on four verticescoincides with W H ( X, T ), where H is a hypergraph with incidence matrix . This can be verified by direct computations; see §9.2. We regard examples such as theabove as evidence that the existence of a toric isomorphism in Theorem 7.1 is perhaps amore natural question to investigate than coincidence of rational functions.69 Graph operations and ask zeta functions of cographs Likewise, the conclusion of Theorem D does not hold for arbitrary graphs: Example 7.5. Let Γ be the graphBy an explicit computation using §6 (see §9.1), we find that W − Γ ( X, T ) = ( − X T + 5 X T − X T + 4 X T − XT + 14 XT − XT + 4 XT + T − T + 5 T + 5 T − T + 1 + 4 X − T − X − T + 14 X − T − X − T + 4 X − T − X − T + 5 X − T − X − T ) / ((1 − T )(1 − XT ) (1 − X T )) , (7.3)where the numerator and denominator are both factored into irreducibles in Q ( X )[ T ].In view of the quadratic irreducible factor 1 − X T in (7.3), Theorem C shows that W − Γ ( X, T ) is not of the form W H ( X, T ) for any hypergraph H . Let Γ be a cograph with vertex set V . At the heart of our constructive proof ofTheorem 7.1 lies the notion of a scaffold on V over a cone σ ⊂ R (cid:62) V ; see Definition 7.7.Informally, scaffolds are forests (i.e. disjoint unions of trees) on V with the same connectedcomponents as Γ. These forests all come with outgoing orientations given by specifyinga root in each of the forest’s trees and letting all edges point away from their associatedroot. Crucially, these orientations are required to be compatible with the preorder (cid:54) σ on Z V induced by the cone σ ; see §4.1. By shrinking σ , we may further assume thatthe restriction of this preorder to V is total, i.e. a weak order. In addition to the above,the edges of a scaffold carry weights in the form of subsets of V . In this way, scaffoldsgive rise to hypergraphs and also to weighted signed multigraphs ( WSM s; see §6.1) andadjacency modules.We say that a scaffold encloses a (co)graph Γ over σ if the adjacency module of the WSM associated with the scaffold and the adjacency module of Γ (with respect to σ )coincide in a strong sense. In this case, we call the scaffold’s hypergraph a local model ofΓ over σ ; see Definition 7.14(i).A fundamental idea behind our proof of Theorem 7.1 is to approximate the graph Γby scaffolds attached to various cones. These cones cover the positive orthant R (cid:62) V .Crucially, all scaffolds realise the same (!) hypergraph H (up to suitable identifications)as local models of Γ. In this case, we call H a global model of Γ; see Definition 7.14(ii).As cographs (save for singletons) arise as either disjoint unions or joins of smallercographs, we are looking to recursively construct (global) models of disjoint unions andjoins of cographs. The case of disjoint unions is comparatively simple: Proposition 7.2370 Graph operations and ask zeta functions of cographs establishes that if Γ and Γ are cographs with modelling hypergraphs H and H , thenthe disjoint union H ⊕ H is a model of Γ ⊕ Γ .The case of joins of (co)graphs, which is settled in Theorem 7.24, is much more involved.In §7.7, we construct a model for the join of Γ ∨ Γ of Γ and Γ by implementing thefollowing strategy.We fix a cone σ ⊂ R (cid:62) V and scaffolds S i ( σ ) enclosing Γ i for i = 1 , 2. In Phase 1 ,using a process governed by removing, one at a time, suitably chosen connecting edgesbetween Γ and Γ , we modify the disjoint union of the scaffolds S i ( σ ) to obtain a scaffold S ( N ) which “almost encloses” the join Γ ∨ Γ ; more precisely, it encloses said join up tofactoring out a particular submodule. It then remains to consider this “error term”.The scaffold S ( N ) differs from the disjoint union of the scaffolds S ( σ ) and S ( σ ) onlyin the weights borne by its edges. The disjoint union of two scaffolds is, in particular, adisjoint union of two forests. In order to obtain a scaffold enclosing the connected (!)graph Γ ∨ Γ , we grow, in Phase 2 of our construction, a single oriented tree out ofthe two oriented forests comprising S ( N ) . In order to ensure that the resulting scaffold S ( ∞ ) has the desired property of giving rise to a local model of Γ ∨ Γ over σ , wegraft judiciously chosen (directed and weighted) edges between pairs of roots from bothforests. A final analysis shows that the given procedure is sufficiently independent ofthe many choices made along the way and, crucially, the chosen cone σ . In particular,the hypergraph associated with S ( ∞ ) essentially only depends on the graphs Γ and Γ and the hypergraphs associated with the scaffolds S ( σ ) and S ( σ ). This allows us tocombine global models of each of Γ and Γ into a global model of the join Γ ∨ Γ . By an orientation of a graph Γ = ( V, E, | · | ), we mean a function ori: E → V × V whichassigns an ordered pair ( u, v ) = ori( e ) to each edge e ∈ E with | e | = { u, v } . We call u and v the source and target of e , respectively. We use the notation u e −→ v for anoriented edge e with ori( e ) = ( u, v ).The indegree (resp. outdegree ) indeg( u ) (resp. outdeg( v )) of u ∈ V with respect toan orientation is the number of edges with target (resp. source) u . An orientation of Γ is outgoing if each vertex has indegree at most one.If T is a tree and u is a vertex of T , then the rooted orientation of T with root u hasall edges pointing away from u . More formally, let e be any edge with | e | = { v, w } , where v precedes w on the unique simple path from u to w . We then define v to be the sourceof e . This orientation of T is clearly outgoing. Trees endowed with such orientations areoften referred to as arborescences or out-trees in the literature. Outgoing and rootedorientations of trees are identical concepts: Proposition 7.6 (Cf. [31, §3.5]) . Let T be a tree endowed with an outgoing orientation ori . Then T contains a vertex u such that ori is the rooted orientation of T with root u .Proof. Let n be the number of vertices of T . Then T contains precisely n − n − 1. Since ori is an outgoing orientation, weconclude that a unique vertex u has indegree zero; see [35, Theorem 16.4]. Let v , . . . , v m Graph operations and ask zeta functions of cographs be the distinct neighbours of u . Let T , . . . , T m be the different trees that constitutethe forest obtained from T by deleting u ; we assume that T i contains v i . Then each T i inherits an outgoing orientation from T . Moreover, v i is the unique vertex in T i withindegree zero. By induction, the induced orientation of each T i is therefore the rootedorientation with respect to v i . The claim for T then follows immediately. (cid:7) In particular, each outgoing orientation of a forest Φ naturally induces a partial order ≺ on V(Φ). In detail, vertices u, v ∈ V(Φ) are comparable if and only if they belong tothe same connected component, C say, and in that case, u ≺ v if and only if u precedes v on the unique simple path from the root of C to v . The ≺ -minimal elements of V(Φ) areexactly the roots of its connected components. Definition 7.7. A scaffold S = (Φ , σ, ori , k · k ) on the vertex set V over a cone σ ⊂ R (cid:62) V consists of a forest Φ = ( V, E, | · | ) endowed with an outgoing orientation ori: E → V × V and a support function k · k : E → P ( V ) such that the following conditions aresatisfied:( S1 ) For each oriented edge u e −→ v in Φ, we have u (cid:54) σ v (see §4.1).( S2 ) k e k 6 = ∅ for each e ∈ E .Given a scaffold S as in Definition 7.7, we obtain a hypergraph H ( S ) := ( V, E, k · k );note that E( H ( S )) = E(Φ) = E . Apart from the outgoing orientation, a scaffold consistsof a forest Φ and a hypergraph H ( S ) related by a common set of (hyper)edges. Ascaffold S as above also gives rise to a weighted signed multigraph (see Definition 6.1) Γ ( S ) := (Γ( S ) , σ, wt S , − S ) := ( V, E( S ) , | · | S ), whereE( S ) := { ( e, w ) : e ∈ E(Φ) , w ∈ k e k} and | ( e, w ) | S := | e | for each ( e, w ) ∈ E( S ).In other words, Γ( S ) is obtained from the forest Φ by replacing each edge e in Φby a set of parallel edges with the same support as e , one for each element of k e k .By condition ( S2 ), the forest Φ and multigraph Γ( S ) determine one another.(ii) The weight of an edge ( e, w ) of Γ( S ) for an oriented (!) edge u e −→ v of Φ and w ∈ k e k is given by wt S ( e, w ) := w − u .Note that u + wt S ( e, w ) = w and v + wt S ( e, w ) = v + w − u both belong to σ ∗ (thelatter since u (cid:54) σ v by ( S1 )) so that condition ( W3 ) in Definition 6.1 is satisfied.72 Graph operations and ask zeta functions of cographs For a scaffold S , we write adj ( S ; R ) := adj ( Γ ( S ); R ) and Adj( S ; R ) := Adj( Γ ( S ); R );as before, we often drop R when R = Z . By definition, for each ring R , adj ( S ; R ) = D X w v − X v + w − u u : u e −→ v in Φ and w ∈ k e k E (cid:54) R σ V. (7.4)Lemma 7.2 provides a sufficient condition for equality of ask zeta functions of adjacencyand incidence representations. In order to use this lemma, we need to be able to establish“toric isomorphisms” between suitable adjacency and incidence modules. Scaffolds providenatural examples of such isomorphisms: Proposition 7.8. Let S be a scaffold as in Definition 7.7. Let C be the set of connectedcomponents of Φ . Then Adj( S ; R ) ≈ R σ Inc( H ( S ) , σ ; R ) ⊕ R σ C .Proof. By a weak scaffold on V , we mean a quadruple S as in Definition 7.7 except thatΦ is allowed to be a forest on a subset V := V(Φ) of V . (This notion will not be usedelsewhere.) The hypergraph H ( S ) associated with a weak scaffold has vertex set V . Wedefine adj ( S ; R ) (cid:54) R σ V as in (7.4) and Adj( S ; R ) := R σ V / adj ( S ; R ).We will establish Proposition 7.8 for weak scaffolds on V by induction on the numberof edges of Φ; our proof uses the same idea as in Lemma 6.11. First, if Φ contains noedges, then | C | = | V | , Adj( S ; R ) = R σ V ≈ R σ R σ C , and Inc( H ( S ) , σ ; R ) = 0.Now suppose that Φ contains at least one edge. Consider any connected component T (a tree!) of Φ consisting of more than one vertex. Then T contains a vertex v withindegree 1 but outdegree 0 (i.e. a leaf distinct from the root). Let u e −→ v be the uniqueoriented edge with target v in T (and in Φ). The change of coordinates on R σ V given by x ( v + X v − u u, if x = v,x, if x ∈ V \ { v } maps the defining generator of adj ( S ; R ) corresponding to e and w ∈ k e k on the right-handside of (7.4) to X w v while preserving generators arising from the other edges (as all thesehave trivial v -coordinate). Let S be the weak scaffold on V obtained by deleting v and e from Φ; the underlying forest Φ of S satisfies V(Φ ) = V \ { v } . Note thatAdj( S ; R ) ≈ Adj( S ; R ) ⊕ R σ / h X w : w ∈ k e ki . Since v is a leaf of T but not a root, deleting it did not increase the number of con-nected components; hence, Φ has precisely | C | connected components. Therefore,Adj( S ; R ) ≈ R σ Inc( H ( S ); R ) ⊕ R σ C by induction and the claim follows using (3.10). (cid:7) Proposition 7.8 allows us to compare adjacency and incidence modules associated withscaffolds. Our next step is to relate the latter to adjacency modules of general graphs. Definition 7.9. Let Γ be a simple graph and S = (Φ , σ, ori , k · k ) be a scaffold as inDefinition 7.7, both on the same vertex set V = V(Γ) = V(Φ). We say that the scaffold S encloses Γ (over the underlying cone σ of S ) if the following conditions are satisfied:73 Graph operations and ask zeta functions of cographs (i) Γ and the underlying forest Φ of S have the same (vertex sets of) connectedcomponents.(ii) adj (Γ , σ ) = adj ( S ). Example 7.10 (Scaffolds and discrete graphs) . Recall that ∆ n is the discrete graph onthe vertex set V = { , . . . , n } ; see (3.1). Trivially, for each cone σ ⊂ R (cid:62) V , there is aunique scaffold on V with underlying forest ∆ n and this scaffold encloses ∆ n . Example 7.11 (Scaffolds and complete graphs) . Consider the complete graph K n on n vertices; see (3.5). To avoid notational confusion, we also denote its vertices by v , . . . , v n (where v i = i ). Let σ = { x ∈ R n (cid:62) : x (cid:54) x , . . . , x n } . Recall from (3.6) that Star n denotes the star graph on { , . . . , n } with centre v = 1.Let S = (Star n , σ, ori , k · k ) be the scaffold on { , . . . , n } , where (i) the edges of Star n areoriented in the form v → v i and (ii) k e k = { v } for each edge e of Star n . Note that S is indeed a scaffold by our definition of σ . In other words, our orientation of Star n iscompatible with the preorder on vertices induced by σ . Figure 2 depicts the scaffold S and the graph K n for n = 3; edges of the former are labelled by their supports in theassociated hypergraph. v v v { v }{ v } Figure 2: A scaffold enclosing K over the cone x (cid:54) x , x We claim that S encloses K n over σ ; our proof of this fact contains ideas that willfeature in our proof of Theorem 7.1.First note that we may identify Z σ = Z [ X , . . . , X n , X − X , . . . , X − X n ] = Z [ X , X − X , . . . , X − X n ] . Next, adj (K n , σ ) = h X i v j − X j v i : 1 (cid:54) i < j (cid:54) n i and adj ( S ) = h X v j − X j v : 2 (cid:54) j (cid:54) n i . In particular, adj ( S ) ⊂ adj (K n , σ ). To show the reverse inclusion, let 1 < i < j (cid:54) n .74 Graph operations and ask zeta functions of cographs Then, over Z σ and modulo adj ( S ), X i v j − X j v i ≡ ( X i v j − X j v i ) + X − X j ( X v i − X i v )= X i v j − X − X i X j v = X − X i · ( X v j − X j v ) ≡ adj ( S )) . Thus, adj ( S ) = adj (K n , σ ) and S encloses K n over σ . Example 7.12 (Scaffolds and P ) . Consider the path Γ = P on three vertices: v v v In the following, we identify v i = i as in Example 7.11. For i = 1 , , 3, let σ i = n x ∈ R (cid:62) : x i (cid:54) x j for j = 1 , , o . We construct scaffolds enclosing P over σ i for each i = 1 , , 3. For i = 2, let S be thescaffold on { v , v , v } with underlying forest P oriented in the form v ← v → v andwith supports k e k = { v } for both edges e , as depicted in Figure 3. v v v { v } { v } Figure 3: A scaffold enclosing P over the cone x (cid:54) x , x Arguments similar to those in Example 7.11 then show that S encloses Γ over σ . Next,by symmetry, the cases i = 1 and i = 3 are interchangeable; we only consider the former.Let S be the scaffold depicted in Figure 4. Since v v v { v } { v } Figure 4: A scaffold enclosing P over the cone x (cid:54) x , x adj (P , σ ) = h X v − X v | {z } =: f , X v − X v | {z } =: f i = h f , f + X − X f i = D f , X (cid:16) v − X − X v (cid:17)E = adj ( S ) , S encloses P over σ . 75 Graph operations and ask zeta functions of cographs Remark 7.13. Clearly, if S encloses Γ over σ and if σ ⊂ σ is a cone, then by shrinkingthe cone of S , we obtain a scaffold S which encloses Γ over σ . Definition 7.14. Let Γ and H be a (simple) graph and hypergraph, respectively, bothon the same vertex set V .(i) We say that H is a local model of Γ over a cone σ ⊂ R (cid:62) V if there existsa scaffold S on V over σ which encloses Γ (see Definition 7.9) together with abijection E( H ( S )) φ −→ E( H ) such that k e k = | eφ | H for all e ∈ E( H ( S )).(ii) We say that H is a (global) model of Γ is there exists a finite set Σ of cones with S Σ = R (cid:62) V such that H is a local model of Γ over each σ ∈ Σ.In other words, H is a local model of Γ over σ if, up to relabelling of its hyperedges, H “is” the hypergraph H ( S ) (see §7.5) of some scaffold S enclosing Γ over σ . In the case ofa global model, the particular scaffold and the relabelling of hyperedges may vary withthe particular cone but the hypergraph remains fixed. Remark 7.15. By Lemma 6.14 and Remark 7.13, we may equivalently require the setΣ in Definition 7.14(ii) to be a fan of cones. Example 7.16. The discrete graph ∆ n is a model of itself. Example 7.17. The block hypergraph BH n,n − (see (3.2)) is a model of the completegraph K n . To see that, consider the cover R n (cid:62) = n S i =1 σ i , where σ i = { x ∈ R n (cid:62) : x i (cid:54) x j for j = 1 , . . . , n } ;here and in the following, we use the notation from Example 7.11. Let S i be the scaffoldover σ i whose underlying graph is the star graph on 1 , . . . , n with centre i , orientededges of the form i → j for i = j , and all hyperedge supports of H ( S i ) equal to { i } . ByExample 7.11, S i encloses K n over σ i .Let H i be a hypergraph with vertices 1 , . . . , n and n − { i } . Then, up to relabelling of its hyperedges, H ( S i ) coincides with H i which istherefore a local model of K n over σ i .Next, by construction, X i divides each X j in Z σ i . In particular, if we redefine allhyperedge supports of H ( S i ) to be { , . . . , n } instead of { i } , the resulting scaffold stillencloses Γ over σ i . Therefore, up to relabelling of its hyperedges, H ( S i ) coincideswith BH n,n − for each i = 1 , . . . , n . We conclude that BH n,n − is a global model of K n . Example 7.18 (A model of P ) . We now construct a global model of P . We continueto use the notation from Example 7.12. For i = 1 , , 3, let σ i = { x ∈ R n (cid:62) : x i (cid:54) x j for j = 1 , . . . , } . Graph operations and ask zeta functions of cographs Let H i be a hypergraph with vertices 1 , , A i , where the rows areordered naturally and A i is given by A = , A = , A = . We showed in Example 7.12 that H i is a local model of P over σ i for i = 1 , , 3. Let H be a hypergraph with vertices 1 , , A = . By redefining hyperedge supports of scaffolds as in Example 7.17 and using that X i divides each X j in Z σ i , we conclude that H is a global model of P . Lemma 7.19. Let H be a local model of Γ over some cone σ . Let c be the number ofconnected components of Γ . Then | E( H ) | = | V(Γ) | − c .Proof. There exists a scaffold S which encloses Γ over σ . As the underlying forest, Φ say,of S and Γ have the same connected components and since a tree on n vertices contains n − H (= number of edges of Φ) is as stated. (cid:7) Proposition 7.20. Let H be a global model of Γ . Let C be the set of connected componentsof Γ . Then Adj(Γ) and Inc( H ) ⊕ Z C are torically isomorphic.Proof. Let Σ be a finite set of cones in R (cid:62) V with S Σ = R (cid:62) V and such that H is alocal model of Γ over each σ ∈ Σ. By Remark 7.15, we may assume that Σ is a fan. Fix σ ∈ Σ. Then, up to relabelling of E( H ), H is the hypergraph H ( S ) associated with ascaffold S which encloses Γ over σ . Hence, by Proposition 7.8, Adj(Γ , σ ) = Adj( S ) ≈ Z σ Inc( H , σ ) ⊕ Z σ C . (cid:7) Corollary 7.21. If H is a (global) model of Γ , then W − Γ ( X, T ) = W H ( X, T ) .Proof. Combine Lemma 7.2 and Proposition 7.20. (cid:7) Remark 7.22. (i) Example 7.17 and Corollary 7.21 provide a new proof of the identity Z ask so n ( O ) ( T ) = Z ask M n × ( n − ( O ) ( T ) from [57, Proposition 5.11].(ii) Corollary 7.21 shows that the graph in Example 7.5 does not admit a global model.Our definition of models is specifically chosen to allow us to prove Proposition 7.20and its consequence Corollary 7.21 as well as Proposition 7.23 and Theorem 7.24. Proposition 7.23 (Models of disjoint unions of graphs) . Let Γ i = ( V i , E i , | · | i ) be a graph for i = 1 , . Let H i = ( V i , H i , k · k i ) be a model of Γ i .Then H ⊕ H (see §3.1) is a model of Γ ⊕ Γ . Graph operations and ask zeta functions of cographs Proof. We may assume that V ∩ V = ∅ . For i = 1 , 2, there exist a collection ofcones Σ i with S Σ i = R (cid:62) V i and a collection of scaffolds ( S i ( σ i )) σ i ∈ Σ i on V i such thateach S i ( σ i ) encloses Γ i over σ i and such that each H ( S i ( σ i )) coincides with H i up torelabelling of hyperedges. Let V := V t V and Σ := { σ × σ : σ i ∈ Σ i ; i = 1 , } sothat R (cid:62) V = S Σ. For σ i ∈ Σ i ( i = 1 , S ( σ , σ ) be the scaffold on V over σ × σ whose underlying forest is the disjoint union of the underlying forests of S ( σ ) and S ( σ ), whose associated hypergraph is the disjoint union of H ( S ( σ )) and H ( S ( σ )),and whose outgoing orientation is induced by those of the scaffolds S i ( σ i ); note thatconditions ( S1 )–( S2 ) are satisfied here so that S ( σ , σ ) is a scaffold. Next, note that adj (Γ ⊕ Γ , σ × σ ) is generated by (the images of) adj (Γ , σ ) and adj (Γ , σ ). Byconstruction, S ( σ , σ ) thus encloses Γ ⊕ Γ over σ × σ whence the claim follows. (cid:7) While formally similar to the preceding proposition, our next result requires considerablymore work; a proof of the following theorem will be given in §7.7. Theorem 7.24 (Models of joins of graphs) . Let Γ i be a non-empty graph for i = 1 , . Write V i = V(Γ i ) . Let H i = ( V i , H i , k · k i ) be amodel of Γ i . Suppose that V ∩ V = ∅ = H ∩ H . Let c i be the number of connectedcomponents of Γ i . Let f ij ( i = 1 , ; j = 1 , . . . , c i − ) and g be distinct symbols, noneof which belongs to H t H . Define a hypergraph H = ( V, H, k · k ) , where V := V t V , H := H t H t { f ij : i = 1 , j = 1 , . . . , c i − } t { g } , and H k ·k −−→ P ( V ) is defined by k h k := k h k t V , if h ∈ H ,V t k h k , if h ∈ H ,V , if h = f j for j = 1 , . . . , c − ,V , if h = f j for j = 1 , . . . , c − ,V t V , if h = g. Then H is a model of the join Γ ∨ Γ (see §3.1) of Γ and Γ . Remark 7.25. (i) The hypergraph H in Theorem 7.24 may be expressed in terms of complete unionsof hypergraphs as follows. For each i ∈ { , } , let H (cid:3) i = ( H i ) ( ci − ; cf. Defini-tion 5.22(iv). Informally speaking, H i and H (cid:3) i coincide except for the multiplicityof the empty hyperedge; an incidence matrix of H (cid:3) i may be obtained from anincidence matrix of H i by inserting c i − | V i | × ( | V i | − H = ( H (cid:3) (cid:16) H (cid:3) ) ; cf. Definition 5.22(iii).(ii) If A i ∈ M n i × ( n i − c i ) ( Z ) are incidence matrices of H i , then the following (with n = n + n ) is an incidence matrix of H : " A n × ( n − c ) n × ( c − n × ( c − n × n × ( n − c ) A n × ( c − n × ( c − n × ∈ M n × ( n − ( Z ) . Graph operations and ask zeta functions of cographs Corollary 7.26. Every cograph admits a model.Proof. Combine the description of cographs in terms of disjoint unions and joins in §7.1,Example 7.16 (for n = 1), Proposition 7.23, and Theorem 7.24. (cid:7) Proof of Theorem 7.1. Combine Corollary 7.26 and Proposition 7.20. (cid:7) Remark 7.27 (Canonical models) . Let Γ be a cograph represented by a cotree as in §7.1.The uniqueness of cotrees shows that the model, H (Γ) say, of Γ constructed in the proofof Corollary 7.26 is uniquely determined by Γ up to isomorphism of hypergraphs fixingall vertices of the set V(Γ) = V( H (Γ)).The hypergraphs of the form H (Γ) for cographs Γ are rather special. Recall that, byCorollary 7.21, W − Γ ( X, T ) = W H (Γ) ( X, T ). Let n = | V(Γ) | . By Lemma 7.19, | E( H (Γ)) | = n − c , where c is the number of connected components of Γ. In particular, Remark 5.20shows that W H (Γ) ( X, T ) can be written over a denominator which is a product of fewerthan 2 n factors of the form 1 − X A T ; for general hypergraphs on n vertices, we obtainan upper bound of 2 n such factors. For another restriction, X e ∈ E( H (Γ)) k e k H (Γ) = 2 | E(Γ) | ;in particular, the number of non-zero entries in any incidence matrix of H (Γ) is even. Example 7.28 (Example 1.6, part III) . We resume the story begun in Example 1.6. Letthe graph Γ ≈ (K ⊕ K ) ∨ K and hypergraph H be as defined there. As we observed inExample 5.25, H ≈ ( BH , ⊕ BH , ) (cid:16) BH , . By Example 7.17, the block hypergraph BH , (resp. BH , ) is a model of the complete graph K (resp. K ). Therefore, byProposition 7.23, the disjoint union BH , ⊕ BH , is a model of K ⊕ K . By Theorem 7.24(see also Remark 7.25(i)),(( BH , ⊕ BH , ) (cid:16) BH , ) ≈ ( BH , ⊕ BH , ) (cid:16) BH , ≈ H is a model of Γ. By Corollary 7.21, W − Γ ( X, T ) = W H ( X, T ) is thus given by (5.30). Before we turn to the proof of Theorem 7.24 we record, for later use, the effects on modelsof taking disjoint unions and joins with a simple graph on a single vertex.We denote, more precisely, by • = K = ∆ a fixed (simple) graph on one vertex andstudy the effects of the operations Γ (cid:32) Γ ⊕ • and Γ (cid:32) Γ ∨ • . Given a hypergraph H , set H = (cid:0) H (cid:1) = ( H ) and likewise H = (cid:0) H (cid:1) = ( H ) ; cf. Definition 5.22. Proposition 7.29. Let H be a model of Γ .(i) H is a model of Γ ⊕ • ,(ii) Let c be the number of connected components of Γ . Then ( H ( c − ) is a model of Γ ∨ • . Graph operations and ask zeta functions of cographs Proof. This follows from Proposition 7.23 (for ⊕ ) and Theorem 7.24 (for ∨ ). (cid:7) Corollary 7.30. (i) For each graph Γ , we have W Γ ⊕• ( X, T ) = W Γ ( X, XT ) ,(ii) For each cograph Γ , we have W Γ ∨• ( X, T ) = − X − T − XT W Γ ( X, X − T ) .Proof. The assertion in (i) follows from [57, §3.4]. For (ii), combine Proposition 7.29(ii)with (5.26) and (5.28). (cid:7) Remark 7.31. The question whether the assumption in (ii) that Γ be a cograph isunnecessary is generalised in Question 10.1. At this point, there is but one missing piece towards our proof of Theorem 7.1 (andTheorem D), namely Theorem 7.24, whose notation we now adopt. Write Γ := Γ ∨ Γ . Similar to the proof of Proposition 7.23, we obtain acollection Σ of cones with S Σ = R (cid:62) V and, for each σ ∈ Σ and i = 1 , 2, a scaffold S i = S i ( σ ) on V i which encloses Γ i over the image σ i of σ under the projection R (cid:62) V (cid:16) R (cid:62) V i such that H ( S i ( σ )) coincides with H i up to relabelling of hyperedges. Using Lemma 4.1,Lemma 6.14, and Remark 7.13 to modify Σ if necessary, we may further assume that (cid:54) σ (see §4.1) induces a total preorder on V for each σ ∈ Σ.Note that in contrast to our proof of Proposition 7.23, we do not assume that each σ ∈ Σ is of the form σ = σ × σ . However, σ ⊂ σ × σ which allows us to identify Z σ i ⊂ Z σ × σ ⊂ Z σ and also e.g. Z σ i V i ⊂ Z σ V . A fixed cone. Henceforth, let σ ∈ Σ be fixed but arbitrary. It suffices to constructa scaffold S which encloses Γ over σ and whose associated hypergraph H ( S ) coincideswith H (as defined in the statement of Theorem 7.24) up to relabelling hyperedges.Write S i := S i ( σ ) = (Φ i , σ i , ori i , k · k i ) and Φ i = ( V i , E i , | · | i ). We may assume that E ∩ E = ∅ . For u i ∈ V i and v j ∈ V j , we use the suggestive notation u i v j both for( u i , v j ) ∈ V i × V j and, if i = j and u i ∼ v i in Φ i , for the oriented edge u i → v i of Φ i . Strategy. For u, v ∈ V , let [ uv ] := X u v − X v u ∈ Z σ V (so [ uv ] = [ u, v ; − 1] as in §3.3).For A ⊂ V × V , write [ A ] := { [ v v ] : v v ∈ A } . Let S (0) = (Φ (0) , σ, ori (0) , k · k (0) )be the “disjoint union” of S and S as constructed in the proof of Proposition 7.23.This is a scaffold enclosing the disjoint union Γ ⊕ Γ over σ . The underlying forestΦ := Φ (0) = Φ ⊕ Φ satisfies E(Φ (0) ) = E t E with the evident support function.80 Graph operations and ask zeta functions of cographs Let M (0) := V × V . Recall that σ i denotes the image of σ under the projection R (cid:62) V (cid:16) R (cid:62) V i . Since S i encloses Γ i over σ i and we identify Z σ i ⊂ Z σ as above, adj (Γ , σ ) = h adj (Γ , σ ) i + h adj (Γ , σ ) i + h [ M (0) ] i = h adj ( S ) i + h adj ( S ) i + h [ M (0) ] i = adj ( S (0) ) + h [ M (0) ] i Beginning with S (0) and M (0) , in the following, we use graph-theoretic operations toconstruct a finite sequence of scaffolds S ( n ) over σ and sets M ( n ) ⊂ V × V such that adj (Γ , σ ) = adj ( S ( n ) ) + h [ M ( n ) ] i for each n (cid:62) 0. The very last of these will satisfy M ( ∞ ) = ∅ and S ( ∞ ) will encloseΓ over σ . Moreover, by construction the hypergraph H ( S ( ∞ ) ) will coincide with H asdefined in Theorem 7.24 up to relabelling of hyperedges. In order to construct S ( n +1) from S ( n ) , we will employ the following observation basedon the same idea as Lemma 6.8. Recall that we assume throughout that (cid:54) σ induces atotal preorder on V . Lemma 7.32 (“Triangle reduction”) . Let S = (Φ , σ, ori , k · k ) be a scaffold on V . Let u → v be an oriented edge in Φ . Let M ⊂ V × V , let z ∈ V , and suppose that uz, vz ∈ M .Let M := M \ { vz } . Define a scaffold S = (Φ , σ, ori , k · k ) via k h k = ( k h k , if h = uv, k uv k ∪ { z } , if h = uv. Then adj ( S ) + h [ M ] i = adj ( S ) + h [ M ] i .Proof. By condition ( S1 ) in Definition 7.7, we have u (cid:54) σ v . Condition ( S2 ) allows us tochoose a vertex w ∈ k uv k . Define g := X w v − X v + w − u u ∈ adj ( S ) ,g := X z v − X v + z − u u ∈ adj ( S ) , and note that adj ( S ) = adj ( S ) + h g i . Further observe that g + [ vz ] = X v − u [ uz ] . (7.5)We claim that h g, [ uz ] , [ vz ] i = h g, g , [ uz ] i over Z σ . To prove that, we consider twocases. First suppose that w (cid:54) σ z . Then g = X z − w g over Z σ and (7.5) implies that h g, [ uz ] , [ vz ] i = h g, [ uz ] i = h g, g , [ uz ] i . Next, suppose that z (cid:54) σ w . Then g + X w − z [ vz ] = X ( v − u )+( w − z ) [ uz ]81 Graph operations and ask zeta functions of cographs over Z σ whence h g, [ uz ] , [ vz ] i = h [ uz ] , [ vz ] i . Moreover, by (7.5) and since g = X w − z g over Z σ , we have h [ uz ] , [ vz ] i = h g , [ uz ] i = h g, g , [ uz ] i . The claim now follows since adj ( S ) + h [ M ] i = adj ( S ) + h g, [ uz ] , [ vz ] i + h [ M ] i = adj ( S ) + h g, g , [ uz ] i + h [ M ] i = adj ( S ) + h [ M ] i . (cid:7) Remark 7.33. Since [ zu ] = − [ uz ], the preceding lemma remains true if uz is replacedby zu or vz is replaced by zv . Invariants. Let ≺ i be the natural partial order induced on V i by the given orientationori i on Φ i ; see §7.4. Let ≺ := ≺ × ≺ be the product order on V × V . Recall that E i denotes the edge set of Φ i . Suppose that ◦ scaffolds S (0) , . . . , S ( n ) on V , ◦ subsets V × V = M (0) ⊃ M (1) ⊃ · · · ⊃ M ( n ) , and ◦ elements v (0) i , . . . , v ( n − i ∈ V i for i = 1 , ‘ = 1 , . . . , n :( M1 ) S ( ‘ ) = (Φ , σ, ori , k · k ( ‘ ) ).( M2 ) M ( ‘ ) is downward closed with respect to ≺ .( M3 ) M ( ‘ ) contains all ≺ -minimal elements of V × V .( M4 ) v ( ‘ − v ( ‘ − ∈ M ( ‘ − and M ( ‘ ) = M ( ‘ − \ { v ( ‘ − v ( ‘ − } .( M5 ) For i + j = 3, if u i v i ∈ E i , then k u i v i k ( ‘ − ⊂ k u i v i k ( ‘ ) ⊂ k u i v i k ( ‘ − ∪ { v ( ‘ − j } .( M6 ) There exist i ∈ { , } and an edge u i → v ( ‘ − i in Φ i with v ( ‘ − j ∈ k u i v ( ‘ − i k ( ‘ ) for i + j = 3.( M7 ) adj (Γ , σ ) = adj ( S ( ‘ ) ) + h [ M ( ‘ ) ] i .Some comments on these conditions are in order. Formalising the strategy in §7.7.1,condition ( M7 ) asserts that adj (Γ , σ ), the module of primary interest to us, coincideswith adj ( S ( ‘ ) ), except for an error measured by M ( ‘ ) . As outlined earlier, the objectiveof our construction is to eventually eliminate this error term.Condition ( M4 ) states that M ( ‘ ) is obtained from M ( ‘ − by removing a single distin-guished pair v ( ‘ − v ( ‘ − . In our construction, these distinguished pairs will be chosenamong ≺ -maximal elements of M ( ‘ − ; when working with the latter, ( M2 ) will be crucial.Condition ( M1 ) asserts that S ( ‘ ) only differs from S (0) in its support function. Thisis made more precise by ( M5 ) which asserts that if e is any edge in Φ i , then the k · k ( ‘ ) -support of e coincides with its k · k ( ‘ − -support except possibly for the addition of the82 Graph operations and ask zeta functions of cographs distinguished vertex v ( ‘ − j —here, j ∈ { , } is the “index distinct from i ”, capturedsuccinctly by the identity “ i + j = 3”. Note that this does not yet rule out the possibilitythat k e k ( ‘ ) = k e k ( ‘ − for all edges e in Φ. However, by ( M6 ), there is some edge e insome Φ i such that k e k ( ‘ ) = k e k ( ‘ − ∪ { v ( ‘ − j } ; in addition, there exists such an edge e such that v ( ‘ − i , the other vertex incident to the distinguished edge from ( M4 ), is incidentto e . Finally, ( M3 ) will guarantee that after finitely many steps, M ( ‘ ) stabilises at theset of minimal elements of M (0) = V × V ; this will conclude Phase 1. Removing non-minimal pairs. Let R i denote the set of ≺ i -minimal elements of V i ; notethat this is precisely the set of roots of the connected components of Φ i . Clearly, R × R is the set of ≺ -minimal elements of V × V .Suppose that M ( n ) (cid:41) R × R and choose a non-minimal pair v v ∈ M ( n ) \ ( R × R ).Let v v be any ≺ -maximal element of M ( n ) with v v ≺ v v ; note that v v R × R .Define v ( n )1 v ( n )2 := v v and M ( n +1) := M ( n ) \ { v v } ; clearly, M ( n +1) is downward closedand M ( n +1) ⊃ R × R .Next, we construct S ( n +1) . Without loss of generality, suppose that v R . (If both v R and v R , we proceed as in the following.) Let u ∈ V be the (unique) ≺ -predecessor of v . Define k · k ( n +1) : E t E → P ( V ) via k h k ( n +1) := ( k h k ( n ) , if h = u v , k u v k ( n ) ∪ { v } , if h = u v . Let S ( n +1) := (Φ , σ, ori , k · k ( n +1) ). Then ( M1 )–( M6 ) are clearly satisfied for ‘ = n + 1.Since v v ∈ M ( n ) and M ( n ) is downward closed, u v belongs to M ( n ) and also to M ( n +1) . It thus follows from Lemma 7.32 that ( M7 ) is satisfied for ‘ = n + 1. Changing support. Since each M ( n +1) is a proper subset of M ( n ) and both of these aresupersets of R × R , the above construction terminates after finitely many steps when M ( N ) = R × R for some N (cid:62) 0. A key property of S ( N ) is the following: Lemma 7.34. Let i + j = 3 . Then for each oriented edge u i → v i in Φ i , k u i v i k (0) ∪ R j ⊂ k u i v i k ( N ) ⊂ k u i v i k (0) ∪ V j . Proof. The second inclusion is immediate from ( M5 ). To prove the first inclusion, weassume, without loss of generality, that i = 1 and j = 2. (When i = 2 and j = 1,we only need to suitably reverse ordered pairs in the following.) Let r ∈ R bearbitrary. It suffices to show that r ∈ k u v k ( N ) . Since v R (because u ≺ v ) and M ( N ) = R × R , we have v r ∈ M (0) \ M ( N ) . Hence, by ( M4 ), for some ‘ ∈ { , . . . , N } ,we have v r = v ( ‘ − v ( ‘ − . As r is a root (= ≺ -minimal element) of one of theconnected components of Φ , the edge u i → v ( ‘ − i in ( M6 ) has to be the given edge u → v . In particular, ( M5 )–( M6 ) imply that r = v ( ‘ − ∈ k u v k ( ‘ ) ⊂ k u v k ( N ) . (cid:7) To proceed further, we need another lemma.83 Graph operations and ask zeta functions of cographs Lemma 7.35 (“Support addition”) . Let S = (Φ , σ, ori , k · k ) be a scaffold on V . Let u → v be an oriented edge of Φ . Let w ∈ k uv k , let z ∈ V , and suppose that w (cid:54) σ z .Define a scaffold S = (Φ , σ, ori , k · k ) via k h k = ( k h k , h = uv, k uv k ∪ { z } , h = uv. Then adj ( S ) = adj ( S ) .Proof. Define g := X w v − X v + w − u u ∈ adj ( S ) ,g := X z v − X v + z − u u ∈ adj ( S )so that adj ( S ) = adj ( S ) + h g i . By assumption, g = X z − w g ∈ adj ( S ) over Z σ . (cid:7) Define k · k ( N +1) : E t E → P ( V ) via k h k ( N +1) := ( k h k t V , if h ∈ E ,V t k h k , if h ∈ E . Let S ( N +1) := (Φ , σ, ori , k · k ( N +1) ) and M ( N +1) := M ( N ) = R × R . Corollary 7.36. adj (Γ , σ ) = adj (cid:16) S ( N +1) (cid:17) + h [ M ( N +1) ] i .Proof. By ( M7 ), it suffices to show that adj ( S ( N ) ) = adj ( S ( N +1) ). Let i + j = 3 andlet h be any oriented edge of Φ i . Let z j ∈ V j be arbitrary. Let r j ∈ R j be the root ofthe connected component of Φ j which contains z j . By condition ( S1 ) in Definition 7.7, r j (cid:54) σ z j . By Lemmas 7.34–7.35, adj ( S ( N ) ) remains unchanged after adding z j to k h k ( N ) .Repeated application gives the desired result. (cid:7) Γ ∨ Γ . By assumption, there exists v ∈ V = V t V such that v (cid:54) σ u for all u ∈ V . Without loss of generality, suppose that v ∈ V . Let a be the root of the connected component of Φ which contains v . By condition ( S1 ) inDefinition 7.7, a (cid:54) σ v so that a too is a (cid:54) σ -minimum of V . Choose b ∈ R amongthe (cid:54) σ -minima of R . Define S ( ∞ ) := (Φ ( ∞ ) , σ, ori ( ∞ ) , k · k ( ∞ ) ) as follows: • Φ ( ∞ ) is the tree (!) with orientation ori ( ∞ ) obtained from Φ = Φ ⊕ Φ by insertinga directed edge a → r i for each r i ∈ ( R \ { a } ) ∪ R . Note that this orientationis outgoing with a as the root of Φ ( ∞ ) . • k · k ( ∞ ) : E(Φ ( ∞ ) ) → P ( V ) is defined via k h k ( ∞ ) = V , if h = a r for r ∈ R \ { b } ,V t V , if h = a b ,V , if h = a r for r ∈ R \ { a } , k h k ( N +1) , otherwise . Graph operations and ask zeta functions of cographs By our choice of a as a (cid:54) σ -minimal element of V , we see that S ( ∞ ) is a scaffold on V . Lemma 7.37. S ( ∞ ) encloses Γ = Γ ∨ Γ over σ .Proof. First note that Φ ( ∞ ) and Γ are both connected: the former by construction andthe latter since it is a join of non-empty graphs. It thus only remains to show that adj (Γ , σ ) = adj ( S ( ∞ ) ). Let F := D X w − a [ a , r ] : r ∈ ( R \ { a } ) t R , w ∈ k a r k ( ∞ ) E (cid:54) Z σ V and note that, by (7.4), adj ( S ( ∞ ) ) = adj ( S ( N +1) ) + F .By condition ( S1 ) in Definition 7.7 and since R i consists of the roots of Φ i , for each v i ∈ V i , there exists r i ∈ R i with r i (cid:54) σ v i . Moreover, a (cid:54) σ v for each v ∈ V and b (cid:54) σ v for each v ∈ V by our choices of a and b . Hence, by the definition of k · k ( ∞ ) , F = D X b − a [ a , r ] : r ∈ R \ { a } E + D [ a , r ] : r ∈ R E . On the other hand, setting G := h [ R × R ] i (cid:54) Z σ V , by Corollary 7.36, adj (Γ , σ ) = adj ( S ( N +1) ) + G . It thus suffices to show that F = G . Write H := h [ a , r ] : r ∈ R i ⊂ F ∩ G . For r ∈ R \ { a } and r ∈ R , since a (cid:54) σ r and a (cid:54) σ r , we obtain the“triangle identity” (cf. Lemma 6.8)[ r , r ] = X r − a [ a , r ] − X r − a [ a , r ] . (7.6)As [ a , r ] ∈ H ⊂ F and X r − a [ a , r ] = X r − b · X b − a [ a , r ] ∈ F , we obtain G ⊂ F .Conversely, by taking r = b in (7.6), we see that X b − a [ a , r ] ∈ G whence F ⊂ G . (cid:7) Remark 7.38. The proof of Lemma 7.37 rested on the validity of the following conditions: • a ∈ k a r k ( ∞ ) for all r ∈ R . • b ∈ k a r k ( ∞ ) ⊂ V for all r ∈ R \ { a } .In particular, numerous alternative definitions of k · k ( ∞ ) are possible while maintainingthe validity of Lemma 7.37. The crucial point of the definition that we chose—tobe exploited in the upcoming final step of our proof of Theorem 7.24—is that, up torelabelling hyperedges, the specific choice that we made works uniformly in all possiblecases. That is to say, it works uniformly for all possible choices of a and b and also inthe case that all (cid:54) σ elements of V belong to V (in which case we choose a ∈ V and b ∈ V and proceed analogously to what we did above). Finale. Recall that c i denotes the number of connected components of Γ i . Note that c i = | R i | by condition ( S2 ) in Definition 7.7. Hence, by unravelling the definition of k · k ( ∞ ) from above, we see that, up to relabelling of hyperedges, the hypergraph H ( S ( ∞ ) )coincides with H in Theorem 7.24. In particular, H is a local model of Γ over our fixedbut arbitrary cone σ from the beginning of this section. This completes the proof ofTheorem 7.24. (cid:7) Cographs, hypergraphs, and cographical groups As in §7, all graphs in this section are assumed to be simple. The story so far. In §3.4 we attached a unipotent group scheme (“graphical groupscheme”) G Γ to each graph Γ. For each compact DVR O we expressed, in Corollary B,the class counting zeta functions of the group scheme G Γ ⊗ O in terms of the rationalfunction W − Γ ( X, T ) from Theorem A(ii): ζ cc G Γ ⊗ O ( s ) = W − Γ (cid:16) q, q | E(Γ) |− s (cid:17) . For a cograph Γ, the Cograph Modelling Theorem (Theorem D) established that thereexists an explicit modelling hypergraph H = H (Γ) for Γ. This is a specific hypergraph onthe same vertex set as Γ which satisfies W − Γ ( X, T ) = W H ( X, T ) , where W H ( X, T ) is the rational function associated with H in Theorem A(i).Our proof of the Cograph Modelling Theorem was constructive. Indeed, cographs (savefor isolated vertices) are disjoint unions or joins of smaller cographs. Given modellinghypergraphs of two cographs we constructed, in Proposition 7.23 and Theorem 7.24,modelling hypergraphs of their disjoint union and join, respectively; cf. Remark 7.27.In §5, we carried out an extensive analysis of the rational functions W H ( X, T ) associatedwith hypergraphs H resulting, in particular, in an explicit formula, viz. Theorem C. Wealso investigated the effects of taking disjoint unions and complete unions of hypergraphs.This ties in well with our constructive proof of the Cograph Modelling Theorem. Namely,by Proposition 7.23, the disjoint union H ⊕ H of modelling hypergraphs H and H of cographs Γ and Γ is a model of the cograph Γ ⊕ Γ . Moreover, the modellinghypergraph of the join Γ ∨ Γ can be described in terms of the complete union H (cid:16) H and the operations from §5.4; cf. Remark 7.25(i).In the present section, we apply the results from §5 to class counting zeta functions ofcographical group schemes. Proof of Theorem E. Let V be the set of vertices of the cograph Γ. Let H = H (Γ) be amodelling hypergraph for Γ with hyperedge multiplicities ( µ I ) I ⊂ V as in Theorems C–D.Our proof of Theorem D in §7 shows that we may assume that P I µ I = n − c , where n and c are the numbers of vertices and connected components of Γ, respectively; cf.Lemma 7.19. Let m be the number of edges of Γ. The bound m (cid:62) n − c then impliesthat for each summand of W H ( X, X m T ) in (1.3), the coefficients of T k in X − (cid:7) Proof of Theorem F. For the first part and the integrality of local poles, combine Corol-lary B, Theorem D, and Theorem 5.26. It remains to prove that the real parts of the86 Cographs, hypergraphs, and cographical groups poles of ζ cc G Γ ⊗ O K ( s ) are positive. Let Γ, V , H , ( µ I ) I ⊂ V , m , and n be as in the proof ofTheorem E above. As we argued there, m − P I ∩ J = ∅ µ I (cid:62) J ⊂ V . In particular, f ( J ) := | J | + m − P I ∩ J = ∅ µ I > J = ∅ . Unless Γ is discrete, f ( ∅ ) = m > n is discrete, then the real parts of the poles of ζ cc G ∆ n ⊗ O ( s ) = 1 / (1 − q n − s ) areequal to n which is positive since cographs are non-empty. (cid:7) Much as for hypergraphs in §5.2, for arbitrary graphs Γ and Γ , the rational function W ± Γ ⊕ Γ ( X, T ) is the Hadamard product of W ± Γ ( X, T ) and W ± Γ ( X, T ). In particular, if R is the ring of integers of a number field or a compact DVR , then the class countingzeta function ζ cc G Γ1 ⊕ Γ2 ⊗ R ( s ) is the Hadamard product of the Dirichlet series ζ cc G Γ1 ⊗ R ( s )and ζ cc G Γ2 ⊗ R ( s ); this simply reflects the fact that class numbers of finite groups aremultiplicative: k( H × H ) = k( H ) × k( H ) for finite groups H and H . -nilpotent groups We now apply §5.2.1 to study class counting zeta functions of cographical groups modelledby hypergraphs with disjoint supports.Let n = ( n , . . . , n r ) ∈ N r . We write n = P ri =1 n i and (cid:0) n (cid:1) = P ri =1 (cid:0) n i (cid:1) . We considerthe cographical group scheme associated with the cographK n = K n ⊕ . . . ⊕ K n r ;see (3.5) and note that K n has m = (cid:0) n (cid:1) edges. These cographs are of specific group-theoretic interest since G K n ( Z ) = F ,n × . . . × F ,n r is the direct product of the freeclass-2-nilpotent groups on n i generators; in particular, G K n ( Z ) = F ,n . Remark 8.1. Conflicting notation in the literature notwithstanding, the cograph K n isnot to be confused with the complete multipartite graph on disjoint, independent sets ofcardinalities n , . . . , n r ; the latter graph will feature as ∆ n in §8.3.1.Combining Proposition 7.23 and Example 7.17, we see that K n is modelled by thehypergraph H (K n ) = BH n , n − = L ri =1 BH n i ,n i − . By Corollary 5.14 (noting that n i − m i = n i − ( n i − 1) = 1 for all i ∈ [ r ]), W − K n ( X, T ) = W H (K n ) ( X, T )= W BH n , n − ( X, T )= X y ∈ d WO r Y i ∈ sup( y ) (1 − X − n i ) Y J ∈ y gp (cid:16) X | J | T (cid:17) . Cographs, hypergraphs, and cographical groups Corollary 8.2. Let n = ( n , . . . , n r ) ∈ N r . For each compact DVR O , ζ cc G K n ⊗ O ( s ) = W − K n ( q, q ( n ) − s ) = X y ∈ d WO r Y i ∈ sup( y ) ( n i ) Y J ∈ y gp (cid:16) q ( n ) + | J |− s (cid:17) . (cid:7) Example 8.3. (i) If r = 1 and n = ( n ), then (cid:0) n (cid:1) = (cid:0) n (cid:1) , whence ζ cc G K n ⊗ O ( s ) = 1 − q ( n − ) − s (cid:16) − q ( n ) − s (cid:17) (cid:16) − q ( n ) +1 − s (cid:17) , in accordance with [47, Corollary 1.5]. There G K n goes by the name F n ,δ , where n = 2n + δ with δ ∈ { , } . See Example 8.23 for a bivariate version of this formula.(ii) If r = 2 and n = ( n , n ), then n = n + n and (cid:0) n (cid:1) = (cid:0) n (cid:1) + (cid:0) n (cid:1) , whence ζ cc G K( n ,n ⊗ O ( s ) = ζ cc G K n ⊗ O ( s ) ? ζ cc G K n ⊗ O ( s ) =1 + q ( n ) +1 − s (cid:0) − q − n − q − n − q − n +1 − q − n +1 + q − n +1 (cid:1) + q ( n ) − n +3 − s (cid:16) − q ( n ) − s (cid:17) (cid:16) − q ( n ) +1 − s (cid:17) (cid:16) − q ( n ) +2 − s (cid:17) . (8.1) -nilpotent products ofcographical groups The results of §5.3 on complete unions of hypergraphs have direct corollaries pertainingto class counting zeta functions of joins of graphs. Recall from §3.4 that for graphs Γ and Γ , the graphical group G Γ ∨ Γ ( Z ) is the free class-2-nilpotent product of G Γ ( Z )and G Γ ( Z ). Proposition 8.4. Let Γ and Γ be cographs on n and n vertices, respectively. Then W − Γ ∨ Γ ( X, T ) = ( X − n − n T − W − Γ ( X, X − n T )(1 − X − n T )(1 − X − n T )+ W − Γ ( X, X − n T )(1 − X − n T )(1 − X − n T )) / ((1 − T )(1 − XT )) . (8.2) In particular, if Γ is a cograph, then W − Γ ∨• ( X, T ) = − X − T − XT · W − Γ ( X, X − T ) .Proof. We may assume that V(Γ ) ∩ V(Γ ) = ∅ . By Corollary 7.26, each Γ i admits amodel, H i say. In particular, W − Γ i ( X, T ) = W H i ( X, T ) for i = 1 , H = (cid:0) H (cid:3) (cid:16) H (cid:3) (cid:1) (see Definition 5.22), where H (cid:3) i = ( H i ) ( ci − and c i is the numberof connected components of Γ i . By Theorem 7.24 and Remark 7.25, H is a model of88 Cographs, hypergraphs, and cographical groups Γ ∨ Γ . Hence, by Corollary 7.21, W − Γ ∨ Γ ( X, T ) = W H ( X, T ). By Proposition 5.23(applying (5.29) c resp. c times and (5.28) once), W H ( X, T ) = 1 − X − T − T · W H (cid:16) H ( X, X − T ) . The claim now follows from Corollary 5.17 by substituting n i − m i in (5.21). Thisreflects the fact that the hypergraphs H (cid:3) i are “near squares”: they have n i vertices and atotal number of n i − (cid:7) Let G Γ and G Γ be the cographical group schemes associated with the cographs Γ and Γ . Let Γ i have n i vertices and m i edges. For each compact DVR O , Proposition 8.4now allows us to express ζ cc G Γ1 ∨ Γ2 ⊗ O = W − Γ ∨ Γ ( q, q m + m + n n − s )in terms of ζ cc G Γ1 ⊗ O ( s ) and ζ cc G Γ2 ⊗ O ( s ). As a special case, we record the following. Corollary 8.5. Let Γ be a cograph with n vertices and m edges. Write • = K = ∆ .Then for each compact DVR O , ζ cc G Γ ∨• ⊗ O ( s ) = 1 − q m + n − − s − q m + n +1 − s ζ cc G Γ ⊗ O ( s + 1 − m − n ) . (cid:7) Remark 8.6. Via the functional equations W − Γ ( X − , T − ) = − X n T W − Γ ( X, T ) in Corol-lary 1.4, the numbers n and n in Proposition 8.4—and hence the left-hand side of(8.2)—are already determined by the rational functions W − Γ i ( X, T ). -nilpotent products of abelian groups We now apply the results and formulae developed in §5.3.1 to cographical groups modelledby hypergraphs with codisjoint supports.As before, let n = ( n , . . . , n r ) ∈ N r , n = P ri =1 n i , and (cid:0) n (cid:1) = P ri =1 (cid:0) n i (cid:1) . We considerthe cographical group scheme associated with the cograph∆ n := ∆ n ∨ . . . ∨ ∆ n r , viz. the complete multipartite graph on disjoint, independent sets of cardinalities n , . . . , n r . Note that ∆ n has m = (cid:0) n (cid:1) − (cid:0) n (cid:1) edges. These cographs are of specific group-theoretic interest since G ∆ n ( Z ) = Z n (cid:16) · · · (cid:16) Z n r (see (3.11)) is the free class-2-nilpotentproduct of free abelian groups of ranks n , . . . , n r . In particular, G ∆ ( r ) ( Z ) = F ,r is thefree class-2-nilpotent group on r generators.By Remark 7.25(ii) and using the notation in Definition 5.22 and (3.4), ∆ n is modelledby H (∆ n ) = ( PH n , n − ) ( r − . By Proposition 5.23 (applying (5.28) r − W H (∆ n ) ( X, T ) = 1 − X − r T − T W PH n , n − ( X, X − r T ) . Cographs, hypergraphs, and cographical groups Combining Corollary B with the explicit formula for W PH n , n − ( X, T ) in Corollary 5.19(substituting n − r for m there), we obtain the following. Proposition 8.7. Let n = ( n , . . . , n r ) ∈ N r and m = (cid:0) n (cid:1) − (cid:0) n (cid:1) . For each com-pact DVR O , ζ cc G ∆ n ⊗ O ( s ) = W − ∆ n ( q, q m − s ) = W H (∆ n ) ( q, q m − s ) = 1 − q − r + m − s − q m − s W PH n , n − ( q, q − r + m − s )= 1(1 − q m − s ) (1 − q m − s ) × (8.3) − q − n + m − s − r X i =1 ( q n i − q n i − − − q n i − n + m − s !! . (cid:7) Example 8.8. Proposition 8.7 unifies and generalises a number of known formulae.(i) If r = 1 and n = ( n ), then m = (cid:0) n (cid:1) − (cid:0) n (cid:1) = 0, confirming the trivial formula ζ cc G ∆ n ⊗ O ( s ) = ζ cc G na ⊗ O ( s ) = 11 − q n − s , where G a denotes the additive group scheme.(ii) If n = ( r ) ∈ N r , then (cid:0) n (cid:1) = 0 = q n i − − 1. Proposition 8.7 thus reconfirms ζ cc G ∆ ( r ) ⊗ O ( s ) = ζ cc G K r ⊗ O ( s ) = 1 − q ( r − ) − s (cid:16) − q ( r ) − s (cid:17) (cid:16) − q ( r ) +1 − s (cid:17) ;see Example 8.3(i).(iii) If r = 2 and n = ( N, N ), then m = (cid:0) n (cid:1) − (cid:0) n (cid:1) = N and 2 n i = n = 2 N , whence ζ cc G ∆ N ∨ ∆ N ⊗ O ( s ) =(1 − q N ( N − − s )(1 − q N ( N − − s ) + q N − s (1 − q − N )(1 − q − N +1 )(1 − q N − s ) (1 − q N − s ) , in accordance with [47, Corollary 1.5], where G ∆ N ∨ ∆ N goes by the name G N . In this section we introduce and study a specially well-behaved class of cographs admitting,in particular, ask zeta functions of “Riemann-type”; see Theorem 8.19. Throughout, • = K = ∆ denotes a fixed simple graph on one vertex. Definition 8.9. A kite graph is any graph belonging to the class Kites which isrecursively defined to be minimal subject to the following conditions:(i) • ∈ Kites . 90 Cographs, hypergraphs, and cographical groups (ii) If Γ ∈ Kites , then • ∨ Γ ∈ Kites and Γ ⊕ • ∈ Kites .(iii) If Γ ∈ Kites and Γ is isomorphic to a graph Γ , then Γ ∈ Kites .As we will see in Theorem 8.19 below, ask zeta functions associated with (negativeadjacency representations of) kite graphs admit a particularly nice and explicit description.Note that every kite graph is a cograph. Further note that any kite graph containsat most one connected component consisting of more than one vertex and that such acomponent is a kite graph itself.Further note that the Z -points of cographical group schemes associated with kite graphsform exactly the class of those torsion-free finitely generated groups of nilpotency class atmost 2 which contains Z and which is closed under taking direct and free class-2-nilpotentproducts with Z . Example 8.10. The following is an example of a connected kite graph:Note that the central vertex is connected to all other vertices. Its removal results in adisconnected graph consisting of one isolated vertex and another component which is astar graph on four vertices. The above graph is therefore isomorphic to (cid:16) (( • ⊕ • ⊕ • ) ∨ • ) ⊕ • (cid:17) ∨ • and is thus a kite graph. Remark 8.11. Neither the so-called Krackhardt kite graph nor the graph on five verticescalled “kite” on [11, p. 18] are kite graphs in the sense of Definition 8.9.We seek to parameterise kite graphs in a useful fashion. Let k , k , . . . be a sequence ofnon-negative integers. Define Kite() to be the empty graph and recursively defineKite( k , . . . , k c +1 ) := ( Kite( k , . . . , k c ) ⊕ ∆ k c +1 , if c is even , Kite( k , . . . , k c ) ∨ K k c +1 , if c is odd . Clearly, Kite( k , . . . , k c ) is a kite graph for each c (cid:62) k , . . . , k c , providedthat at least one k i is positive. (The empty graph is neither a kite graph nor a cograph.) Example 8.12. Kite( n ) = ∆ n and Kite(1 , n − 1) = K n .Recall that a composition of a non-negative integer n is a sequence k = ( k , . . . , k c )of positive integers with n = k + · · · + k c . We tacitly identify compositions and infinitesequences k = ( k , k , . . . ) such that k i = 0 for some i and, in addition, k j = 0 whenever i < j and k i = 0. 91 Cographs, hypergraphs, and cographical groups Proposition 8.13. (i) Every kite graph on n vertices is isomorphic to Kite( k , . . . , k c ) for some composition ( k , . . . , k c ) of n .(ii) Let k and k be compositions of positive integers. Then Kite( k ) and Kite( k ) areisomorphic if and only if k = k .Proof. Given a label ‘ and rooted labelled trees T , . . . , T u , let ( ‘, T , . . . , T u ) denote therooted tree whose root, v say, has label ‘ and such that the descendant trees of v areprecisely the trees T , . . . , T u . For notational convenience, we identify a label ‘ and therooted labelled tree ( ‘ ). We see that kite graphs with vertex set { , . . . , n } are preciselythose cographs with cotrees (see §7.1) of the form * . . . (cid:26) ⊕ , [ ∨ , ( ⊕ , , , . . . , k ) , k + 1 , . . . , k + k ] , k + k + 1 , . . . , k + k + k (cid:27) , . . . + , where ( k , k , . . . ) is a composition of n and we used different types of parentheses forclarity. The uniqueness of cotrees of cographs (see §7.1) now implies both claims. (cid:7) Example 8.14 (Example 8.10, part II) . The kite graph in Example 8.10 is isomorphicto ((∆ ∨ K ) ⊕ ∆ ) ∨ K ≈ Kite(3 , , , . Corollary 8.15. Let n (cid:62) . Up to isomorphism, there are precisely n − kite graphson n vertices. Among these, precisely n − are connected.Proof. By construction, for a composition k = ( k , . . . , k c ), the graph Kite( k ) is connectedif and only if c is even. As is well-known, there are precisely 2 n − compositions of n andit is easily verified that precisely half of these have even length. (cid:7) Let k = ( k , k , . . . ) be a composition of a positive integer. For t (cid:62) 1, let k ( t ) := t P i =1 k i and k [ t ] := ∞ P i = t ( − i +1 k i ; note that k [ t ] = 0 = k t for t (cid:29) 0. The following is easily provedby induction. Lemma 8.16. | V(Kite( k )) | = ∞ X i =1 k i , | E(Kite( k )) | = ∞ X i =1 k (2 i )2 ! − k (2 i − ! . (cid:7) Recall from (3.9) the notion of the staircase hypergraph ΣH m associated with a vector m = ( m , . . . , m n ) ∈ N n +10 . Proposition 8.17. Every kite graph admits a staircase hypergraph as a model. Cographs, hypergraphs, and cographical groups Proof. We proceed by induction on the length c of the composition k = ( k , . . . , k c )representing a given kite graph. For c = 1, note that ΣH (0 ,..., is a model of Kite( k ) =∆ k . Next, supposing that Kite( k , . . . , k r − ) admits a model of the form ΣH m , we obtaina model ΣH m of Kite( k , . . . , k r ) by repeated application of Proposition 7.29. (cid:7) Example 8.18 (Example 8.10, part III) . The staircase hypergraph H := ΣH (0 , , , , , , with incidence matrix is a model of the kite graph Kite(3 , , , 1) in Example 8.10.Combining Proposition 8.17 with Proposition 5.9, we see that the rational function W − Kite( k ) ( X, T ) associated with a kite graph Kite( k ) is of a particularly simple form. Thefollowing theorem, which is the main result of the present section, spells this out. Theorem 8.19. Let k = ( k , k , . . . ) be a composition of a positive integer. Then W − Kite( k ) ( X, T ) = 11 − X k [1] T ∞ Y i =1 (1 − X k [2 i +1] − k i +1 T )(1 − X k [2 i +1] − k i T )(1 − X k [2 i +1]+1 T )(1 − X k [2 i +1] T ) . (8.4) Proof. Straightforward induction along blocks( k , k , . . . , k ρ − , k ρ ) (cid:32) ( k , k , . . . , k ρ +1 , k ρ +2 )using Corollary 7.30. (cid:7) Example 8.20 (Example 8.10, part IV) . Consider the graph Kite( k ) for k = (3 , , , k [1] = 2 and k [3] = k [5] = 0. Theorem 8.19 thus asserts that W − Kite(3 , , , ( X, T ) = 11 − X T · (1 − T )(1 − X − T )(1 − XT )(1 − T ) · (1 − T )(1 − X − T )(1 − XT )(1 − T )= (1 − X − T ) (1 − XT ) (1 − X T ) . For kite graphs, we can strengthen Theorem F. Recall that | E(Kite( k )) | is given byLemma 8.16. Theorem 8.21. Let k = ( k , k , . . . ) be a composition of a positive integer. Then forevery number field K with ring of integers O K , the abscissa of convergence of the classcounting zeta function ζ cc G Kite( k ) ⊗O K ( s ) is equal to α (Kite( k )) = | E(Kite( k )) | + max { k [1] + 1 , k [2 i + 1] + 2 : i ∈ N } . The function ζ cc G Kite( k ) ⊗O K ( s ) may be meromorphically continued to all of C . Cographs, hypergraphs, and cographical groups Proof. By Theorem 8.19, the Euler product ζ cc G Kite( k ) ⊗O K ( s ) = Y v ∈V K W − Kite( k ) ( q v , q | E(Kite( k )) |− sv )is a product of finitely many translates of the Dedekind zeta function ζ K ( s ) and inversesof such translates. The abscissa of convergence is then readily read off from (8.4). (cid:7) Example 8.22 (Example 8.10, part V) . The graphical group scheme G Kite(3 , , , hasthe property that, for each number field K , ζ cc G Kite(3 , , , ⊗O K ( s ) = ζ K ( s − ζ K ( s − ζ K ( s − , with global abscissa of convergence α (Kite(3 , , , { , } , inaccordance with Theorem 8.21. For a finite group G , let cc n ( G ) denote its number of conjugacy classes of size n and let ξ cc G ( s ) := ∞ P n =1 cc n ( G ) n − s be the associated Dirichlet polynomial; note that k( G ) = ξ cc G (0).Let G be a unipotent group scheme over the ring of integers O = O K of a numberfield K ; see [66, §2.1]. For a place v ∈ V K , let P v ∈ Spec( O ) be the associated primeideal with residue field size q v = |O / P v | and let O v = lim ←− k O / P kv .In [46, Definition 1.2], Lins defined the bivariate conjugacy class zeta function Z cc G ⊗O ( s , s ) = X = I/ O ξ cc G ( O /I ) ( s ) · |O /I | − s = Y v ∈V K Z cc G ⊗O v ( s , s ) (8.5)associated with G , where the Euler factors are given by Z cc G ⊗O v ( s , s ) = ∞ X i =0 ξ cc G ( O / P iv ) ( s )( q − s v ) i . (8.6)For all but (possibly) finitely many places v ∈ V K , the Euler factors (8.6) are rationalfunctions in q − s and q − s ; see [46, Theorem 1.2]. Both these local and the global zetafunctions (8.5) refine the class counting zeta functions defined in §1.3. Indeed, as observedin [46, §1.2], Z cc G ⊗O (0 , s ) = ζ cc G ⊗O ( s ) . (8.7)(Lins used slightly different notation for these functions; see Remark 8.24.) Just asunivariate ask zeta functions may be expressed in terms of carefully designed univariate p -adic integrals, Lins expressed bivariate conjugacy zeta functions in terms of suitablydefined bivariate p -adic integrals; see [46, §4].We record here, in all brevity, that expressing class counting zeta functions ζ cc G Γ ⊗ O ( s ) = W H ( q, q | E(Γ) |− s ) associated with a cographical group scheme G Γ in terms of the ask zeta94 Cographs, hypergraphs, and cographical groups function of a modelling hypergraph H is compatible with Lins’s bivariate refinement ofclass counting zeta functions. The reason for this is the common multivariate origin ofall the p -adic integrals involved.To be more precise, let H = ( V, E, | · | ) be a hypergraph with incidence representation η .For each compact DVR O , we define the bivariate ask zeta function ζ ask η O ( s , s ) = (1 − q − ) − Z O V × O | y | ( s +1) | E | + s −| V |− Y e ∈ E k x e ; y k − s − d µ O V × O ( x, y );note that ζ ask η O ( s ) = ζ ask η O (0 , s ); cf. Proposition 3.4. One may define bivariate ask zetafunctions in greater generality but we shall not need this here.Generalising (5.2) (for D = ∅ ), we may express the bivariate ask zeta functions interms of the multivariate function Z V, ∅ ( s ) (see (5.1)): ζ ask η O ( s , s ) = (1 − q − ) − Z V, ∅ (cid:0) ( s + 1) | E | + s − | V | − 1; ( µ I ( − s − I ⊂ V (cid:1) . As in §4.4, there exists a rational function W H ( X, T , T ) ∈ Q ( X, T , T ) such that ζ ask η O ( s , s ) = W H ( X, T , T );of course, W H ( X, T ) = W H ( X, , T ). We may then use the multivariate nature of (5.8)to deduce a “trivariate” analogue of the formula for W H ( X, T ) given in Corollary 5.6.If H is a model of a cograph Γ, then Lins’s bivariate conjugacy class zeta function isrecovered via the formula Z cc G Γ ⊗ O ( s , s ) = W H ( q, q − s , q | E(Γ) |− s );this is based on a trivariate form of Theorem D. Example 8.23. For the block hypergraph BH n,m , we readily obtain W BH n,m ( X, T , T ) = 1 − X − m T m T (1 − T )(1 − X n − m T m T ) , (8.8)generalising the bivariate formula given in Example 5.10(i). Using the fact that BH n,n − is a model of the complete graph K n (see Example 7.17) we may use this formula (with n = 2n + δ and m = n − 1) to recover Lins’s formula (cf. [47, Theorem 1.4]) Z cc F n ,δ ⊗ O ( s , s ) = Z cc G K n ⊗ O ( s , s ) = 1 − q ( n − ) − ( n − s − s (cid:16) − q ( n ) − s (cid:17)(cid:16) − q ( n ) +1 − ( n − s − s (cid:17) . As predicted by (8.7), setting ( s , s ) = (0 , s ), we recover the formula in Example 8.3(i).95 Further examples Remark 8.24. (i) Lins’s notation [46, 47] differs slightly from ours. Her Z cc G ( R ) ( s , s ) is what wecalled Z cc G ⊗ R ( s , s ), for various rings R . Our class counting zeta function ζ cc G ⊗ R ( s )goes by the name class number zeta function ζ k G ( R ) ( s ) in Lins’s work. We note thatour ζ cc G ⊗O ( s ) may not only be obtained by suitably specialising Lins’s bivariateconjugacy class zeta function as in (8.7) but also by specialising her bivariaterepresentation zeta function Z irr G ( O ) ( s , s ). The latter is defined analogouslyto (8.5) by enumerating the ordinary irreducible characters of the finite groups G ( O /I ) by degree (rather than conjugacy classes by cardinality); see [46, (1.2)].As is apparent from our discussion here, the techniques developed and employedin our study of class counting zeta functions of (co)graphical group schemes areslanted towards counting conjugacy classes rather than irreducible characters.(ii) Many of our results about univariate local and global class counting zeta functionsassociated with (co)graphical group schemes have bivariate analogues. The bivariateversion of Theorem 5.26, for instance, describes the domain of convergence of thebivariate ask zeta functions from above. General analytic properties of bivariateconjugacy class and representation zeta functions associated with unipotent groupschemes over number fields are studied in [48].(iii) Beyond cographs, using suitable bivariate versions of Theorem A(ii) and Corollary B,we may strengthen Corollary 1.3 as follows: for each simple graph Γ and k (cid:62) 1, thenumber of conjugacy classes of G Γ ( F q ) of size q k is given by a polynomial in q . In this section, we collect a number of further examples of the function W ± Γ ( X, T ) forgraphs Γ beyond the infinite families covered in §8. Our constructive proof of Theorem A (see §§4.4, 6.4) leads to algorithms for computingthe rational functions W H ( X, T ) and W ± Γ ( X, T ) associated with a hypergraph and graph,respectively, complementing the formulae derived in §5. In detail, given a hypergraph H ,Proposition 3.4 expresses W H ( X, T ) in terms of a combinatorially defined p -adic integral.The latter can be expressed as a univariate specialisation of the integrals studied in [54, §3].In particular, [54, Proposition 3.9] and [58, §6] together provide practical means forcomputing W H ( X, T ). Behind the scenes, these techniques rely on algorithms due toBarvinok and Woods [4] for computing with rational generating functions enumeratinglattice points in polyhedra.Regarding the case of a graph Γ, the inductive proof of Theorem 6.4 in §6.4 readilytranslates into a recursive algorithm for computing W ± Γ ( X, T ). For the base case, combineProposition 6.5 and Proposition 4.8 with [54, §3] and [58, §6] as above.96 Further examples Based on the steps just outlined, the first author’s software package Zeta [61] forthe computer algebra system SageMath [67] includes implementations of algorithms forcomputing the rational functions W H ( X, T ) and W ± Γ ( X, T ) in Theorem A. In practice,these algorithms often substantially outperform the previously existing functionality forcomputing ask zeta functions based on [57] that is available in Zeta ; note, however, thatthe present algorithms are only applicable in the context of ask zeta functions associatedwith graphs and hypergraphs. We used them, for instance, to compute the rationalfunctions W − Γ ( X, T ) for all simple graphs on at most seven vertices.In the remainder of this section, we record a number of explicit examples of thefunctions W H ( X, T ) and W ± Γ ( X, T ) computed with the help of Zeta . Table 1 lists both types of rational functions W ± Γ ( X, T ) for all 18 (isomorphism classesof non-empty) simple graphs on at most four vertices; any entry “%” in the column W +Γ ( X, T ) indicates that W − Γ ( X, T ) = W +Γ ( X, T ) for the specific graph Γ in question. Allgraphs in Table 1, save for the path P , are cographs. In particular, 17 of the formulae inTable 1 could, in principle, be derived from Theorems C–D. We further note that all butthe following graphs in Table 1 are kite graphs: K ⊕ K , P , and C ; see Question 10.5. In this subsection, we list W − Γ ( X, T ) for all 34 simple graphs on five vertices. ByCorollary 7.30(i) and using Table 1, it suffices to consider simple graphs on five verticeswithout isolated vertices; there are precisely 23 of these and their associated rationalfunctions W − Γ ( X, T ) are listed in Table 2. We chose not to include the (often bulky)corresponding rational functions W +Γ ( X, T ). In Table 2, cographs are flagged and kitegraphs are labelled as such. Recall that P n and C n denote the path and cycle graph on n vertices, respectively;see (3.7)–(3.8). The rational functions W − P n ( X, T ) and W − C n ( X, T ) for n (cid:54) n denote the group scheme of upper unitriangular n × n matrices. As wenoted in §1.4, the graphical group scheme G P n is isomorphic to the maximal quotientU n +1 , := U n +1 /γγγ (U n +1 ) of U n +1 of class at most 2. The class numbers of the finitegroups U n +1 , ( F q ) were determined by Marjoram [50, Theorem 7]. For n (cid:54) 9, inaccordance with Corollary B, his general formulae agree with the coefficients of T in theexpansions of the rational functions W − P n ( X, X n − T ) recorded in Table 3. W +(K ⊕ K ) ∨ K We may now finish Example 1.6: Table 5 records the numerator of W +Γ ( X, T ) in (1.7).97 F u r t h e r e x a m p l e s Γ W − Γ ( X, T ) W +Γ ( X, T ) / (1 − XT ) % / (1 − X T ) % − X − T (1 − T )(1 − XT ) % / (1 − X T ) % − T (1 − XT )(1 − X T ) % − X − T (1 − XT ) % 02 3 − X − T (1 − T )(1 − XT ) ( T + T + 1 − X − T − X − T + 6 X − T + 3 X − T − X − T − X − T − X − T ) / (1 − T ) / (1 − X T ) % − XT (1 − X T )(1 − X T ) XT − T +1+ X − T − X − T + X − T (1 − X T )(1 − XT )(1 − T ) % − T (1 − X T ) − XT +3 T − T +1 − X − T + X − T − X − T + X − T (1 − T )(1 − XT ) % T +1 − X − T − X − T + X − T + X − T (1 − T ) (1 − XT ) % (1 − X − T )(1 − X − T )(1 − T ) (1 − XT ) ( T + 2 T + 3 T + 1 − X − T − X − T − X − T + 3 X − T − X − T + 4 X − T + 7 X − T + 3 X − T − X − T − X − T − X − T − X − T ) / ((1 − X − T )(1 − T ) (1 − XT )) 01 2 3 (1 − X − T )(1 − XT )(1 − X T ) ( X T − XT + XT − T + 6 T − T + 1 − X − T + 3 X − T − X − T ) / (1 − XT ) 01 2 3 (1 − X − T ) (1 − T )(1 − XT ) ( − XT − XT + 4 T + 4 T − T + 2 T + 1 − X − T + 2 X − T +2 X − T − X − T + X − T + 2 X − T − X − T + 4 X − T +4 X − T − X − T − X − T ) / ((1 − T ) (1 − XT )(1 − XT )) 012 3 − X − T (1 − XT )(1 − X T ) % 012 3 − X − T (1 − T )(1 − XT ) (1+3 X − T +5 X − T +3 X − T − X − T − X − T − X − T +10 X − T + 14 X − T + 8 X − T − X − T − X − T − X − T − X − T ) / ((1 − X − T )(1 − X − T )(1 − T ) ) Table 1: Graphs on at most four vertices and their ask zeta functions Further examples Table 2: Graphs without isolated vertices on at most five vertices and their negative askzeta functions Γ comment W − Γ ( X, T ) Kite(4 , (1 − X − T )(1 − XT )(1 − X T ) 012 3 4 no cograph XT − T +1+ X − T − X − T + X − T (1 − XT ) (1 − X T ) 012 34 Kite(1 , , , (1 − X − T )(1 − T )(1 − XT ) (1 − X T ) no cograph (P ) ( − XT + XT + 4 T − T + 1 − X − T +2 X − T − X − T − X − T + 2 X − T ) / (1 − XT ) 01 23 4 no cograph − XT +3 T − T +1 − X − T + X − T − X − T + X − T (1 − XT ) 012 34 Kite(2 , , , (1 − X − T ) (1 − XT ) cograph T +1 − X − T − X − T − X − T + X − T + X − T (1 − X − T )(1 − XT ) 01 23 4 Kite(3 , (1 − X − T )(1 − X − T )(1 − T )(1 − XT ) 012 34 no cograph − XT + T +1+3 X − T − X − T − X − T − X − T + X − T (1 − T )(1 − XT ) 012 34 no cograph T +1 − X − T − X − T + X − T + X − T (1 − T )(1 − XT ) 012 34 Kite(1 , , , (1 − X − T )(1 − X − T )(1 − T )(1 − XT ) 01 2 3 4 no cograph − XT + T +1+3 X − T − X − T − X − T − X − T + X − T (1 − T )(1 − XT ) Further examples 012 34 cograph T +1 − X − T − X − T + X − T + X − T (1 − XT ) (1 − T ) no cograph (C ) ( T + 3 T + 1 − X − T − X − T + 5 X − T − X − T + 5 X − T + 5 X − T − X − T − X − T − X − T ) / ((1 − T ) (1 − XT )) no cograph − X − T − X − T +3 X − T − X − T + X − T + X − T − X − T (1 − T ) (1 − XT ) cograph − X − T + X − T − X − T +3 X − T − X − T + X − T − X − T (1 − XT )(1 − X − T )(1 − T ) Kite(1 , , , (1 − X − T ) (1 − T ) (1 − XT ) cograph X − T − X − T − X − T + X − T + X − T (1 − X − T )(1 − T )(1 − XT ) Kite(2 , (1 − X − T )(1 − X − T )(1 − X − T )(1 − T )(1 − XT ) cograph − X T + X T − XT +3 XT − T +1 − X − T + X − T (1 − XT ) (1 − X T ) 02 3 34 cograph XT − T +1 − X − T + X − T − X − T + X − T (1 − T )(1 − XT )(1 − X T ) no cograph ( T + 1 − X − T − X − T + 4 X − T − X − T + 2 X − T + 2 X − T − X − T − X − T ) / ((1 − T ) (1 − XT )) Kite(1 , 4) = K − X − T (1 − T )(1 − XT ) Further examples Γ W − Γ ( X, T )P / (1 − XT )P (1 − X − T ) / ((1 − T )(1 − XT ))P (1 − X − T ) / ((1 − XT ) P ( − XT + 3 T − T + 1 − X − T + X − T − X − T + X − T ) / ((1 − XT ) (1 − T ))P ( − XT + XT +4 T − T +1 − X − T +2 X − T − X − T − X − T +2 X − T ) / (1 − XT ) P ( X T − X T + X T − XT + 11 XT − XT + 3 XT + 7 T − T + 20 T − T + 1 − X − T + 6 X − T − X − T + 12 X − T − X − T − X − T + 13 X − T − X − T + 6 X − T − X − T + X − T − X − T ) / ((1 − T )(1 − XT ) )P ( X T + 3 X T − X T + 7 X T − XT + 41 XT − XT + 7 XT + 9 T − T +45 T − T + 1 − X − T + 12 X − T − X − T + 28 X − T − X − T − X − T +40 X − T − X − T +12 X − T − X − T +12 X − T − X − T − X − T ) / ((1 − XT ) )P ( − X T + X T − X T + 14 X T − X T + 14 X T + 10 X T − X T +200 X T − X T + 31 X T − XT + 185 XT − XT + 462 XT − XT +14 XT + 11 T − T + 310 T − T + 189 T − T + 1 − X − T + 26 X − T − X − T +374 X − T − X − T +76 X − T − X − T − X − T +161 X − T − X − T +496 X − T − X − T +25 X − T − X − T +150 X − T − X − T +84 X − T − X − T − X − T + 28 X − T − X − T + X − T − X − T + X − T ) / ((1 − XT ) (1 − T ))P ( X T − X T + 26 X T − X T + 110 X T − X T + 109 X T + 25 X T − X T + 1162 X T − X T + 109 X T − XT + 559 XT − XT +1962 XT − XT +26 XT +8 T − T +918 T − T +582 T − T +1 − X − T +50 X − T − X − T + 1526 X − T − X − T + 94 X − T − X − T − X − T +486 X − T − X − T + 2042 X − T − X − T + 46 X − T − X − T +798 X − T − X − T + 376 X − T − X − T − X − T + 282 X − T − X − T + 4 X − T − X − T + 16 X − T − X − T ) / ((1 − XT ) ) Table 3: Ask zeta functions associated with paths on at most nine vertices101 Further examples Γ W − Γ ( X, T )C (1 − X − T ) / ((1 − XT )(1 − T ))C ( T + 1 − X − T − X − T + X − T + X − T ) / ((1 − XT )(1 − T ) )C ( T + 3 T + 1 − X − T − X − T + 5 X − T − X − T + 5 X − T + 5 X − T − X − T − X − T − X − T ) / ((1 − XT )(1 − T ) )C ( T + 8 T + 8 T + 1 − X − T − X − T − X − T + 13 X − T + 28 X − T − X − T − X − T + 28 X − T + 13 X − T − X − T − X − T − X − T + X − T +8 X − T + 8 X − T + X − T ) / ((1 − XT )(1 − T ) )C ( T +17 T +41 T +17 T +1 − X − T − X − T − X − T − X − T +21 X − T +189 X − T + 175 X − T − X − T − X − T + 70 X − T + 28 X − T − X − T − X − T − X − T + 35 X − T + 168 X − T + 98 X − T + 7 X − T − X − T − X − T − X − T − X − T − X − T ) / ((1 − XT )(1 − T ) )C ( T +33 T +158 T +158 T +33 T +1 − X − T − X − T − X − T − X − T − X − T + 28 X − T + 660 X − T + 1884 X − T + 860 X − T + 24 X − T − X − T − X − T − X − T − X − T +46 X − T +46 X − T − X − T − X − T − X − T − X − T +24 X − T +860 X − T +1884 X − T +660 X − T +28 X − T − X − T − X − T − X − T − X − T − X − T + X − T + 33 X − T +158 X − T + 158 X − T + 33 X − T + X − T ) / ((1 − XT )(1 − T ) )C ( T +60 T +516 T +1015 T +516 T +60 T +1 − X − T − X − T − X − T − X − T − X − T − X − T + 36 X − T + 1770 X − T + 10974 X − T +13896 X − T + 3603 X − T + 87 X − T − X − T − X − T − X − T − X − T − X − T + 60 X − T + 117 X − T + 2268 X − T + 3456 X − T − X − T − X − T − X − T − X − T + 1287 X − T + 11760 X − T +14154 X − T + 3141 X − T + 84 X − T − X − T − X − T − X − T − X − T − X − T − X − T + 153 X − T + 2358 X − T + 6192 X − T +3798 X − T + 504 X − T + 9 X − T − X − T − X − T − X − T − X − T − X − T − X − T − X − T ) / ((1 − XT )(1 − T ) ) Table 4: Ask zeta functions associated with cycles on at most nine vertices102 Further examples X − T + 18 X − T − X − T + 12 X − T + 4 X − T − X − T + 44 X − T − X − T − X − T − X − T + 83 X − T − X − T + 345 X − T +105 X − T + 28 X − T + 32 X − T − X − T + 707 X − T − X − T +210 X − T + 265 X − T − X − T + 74 X − T + 92 X − T − X − T +1415 X − T − X − T − X − T + 137 X − T − X − T + 18 X − T − X − T + 59 X − T − X − T + 409 X − T − X − T − X − T +1329 X − T − X − T + X − T + 8 X − T + 33 X − T + 1872 X − T − X − T + 2093 X − T + 1360 X − T − X − T + 425 X − T − X − T +93 X − T − X − T − X − T − X − T − X − T +1086 X − T − X − T + 553 X − T − X − T − X − T + 35 X − T − X − T +2585 X − T +806 X − T +741 X − T +1094 X − T − X − T +117 X − T − X − T + 56 X − T + 379 X − T + 2070 X − T − X − T + 268 X − T − X − T − X − T + 1093 X − T − X − T + 10 X − T − X − T − X − T − X − T − X − T − X − T + 88 X − T + 3403 X − T − X − T + 1253 X − T + 74 X − T + 37 X − T + 688 X − T + 2069 X − T +2381 X − T + 1478 X − T − X − T + 401 X − T − X − T − X − T − X − T + 26 X − T + 315 X − T − X − T − X − T − X − T − X − T − X − T + 974 X − T + 428 X − T +120 X − T − X − T − X − T + 667 X − T + 529 X − T + 1568 X − T +4496 X − T +1568 X − T +529 X − T +667 X − T − X − T − X − T +120 X − T + 428 X − T + 974 X − T − X − T − X − T − X − T − X − T − X − T +315 X − T +26 X − T − X − T − X − T − X − T + 401 X − T − X − T + 1478 X − T + 2381 X − T +2069 X − T + 688 X − T + 37 X − T + 74 X − T + 1253 X − T − X − T +3403 X − T + 88 X − T − X − T − X − T − X − T − X − T − X − T + 10 X − T − X − T + 1093 X − T − X − T − X − T + 268 X − T − X − T + 2070 X − T + 379 X − T +56 X − T − X − T + 117 X − T − X − T + 1094 X − T + 741 X − T +806 X − T + 2585 X − T − X − T + 35 X − T − X − T − X − T +553 X − T − X − T + 1086 X − T − X − T − X − T − X − T − X − T + 93 X − T − X − T + 425 X − T − X − T +1360 X − T +2093 X − T − X − T +1872 X − T +33 X − T +8 X − T + X − T − X − T + 1329 X − T − X − T − X − T + 409 X − T − X − T + 59 X − T − X − T + 18 X − T − X − T + 137 X − T − X − T − X − T +1415 X − T − X − T +92 X − T +74 X − T − X − T +265 X − T +210 X − T − X − T +707 X − T − X − T +32 X − T + 28 X − T + 105 X − T + 345 X − T − X − T + 83 X − T − X − T − X − T − X − T + 44 X − T − X − T + 4 X − T +12 X − T − X − T + 18 X − T + 5 X − T + X − T Table 5: Numerator F ( X, T ) of W +(K ⊕ K ) ∨ K in (1.7)103 10 Open problems Inspired by the theoretical results and the explicit formulae in this article, we raise anumber of further questions (beyond Questions 1.7 and 1.8) pertaining to the topicscovered here. As we mentioned in §8.2, the effect of taking disjoint unions of graphs correspondsto taking Hadamard products of the rational functions W ± Γ ( X, T ). In particular, forarbitrary simple graphs Γ and Γ , the rational function W − Γ ⊕ Γ ( X, T ) does not dependon the individual graphs Γ and Γ but only on the rational functions attached to these.In the special case that Γ and Γ are both cographs , Proposition 8.4 similarly expresses W − Γ ∨ Γ ( X, T ) in terms of the W − Γ i ( X, T ). As we mentioned in Remark 8.6, even thoughthe formula in Proposition 8.4 involves the numbers of vertices of Γ and Γ , thesenumbers can be recovered from the corresponding rational functions via the functionalequation in Corollary 1.4. Question 10.1 (Joins) . Do the conclusions of Proposition 8.4 hold for all simple graphs?The smallest graph which is not a cograph is the path P on four vertices; the rationalfunction W − P ( X, T ) is recorded in Table 1 (and also in Table 3). Using Zeta , we find that W − P ∨ P ( X, T ) = (1 + 2 X − T − X − T − X − T − X − T + 2 X − T − X − T + X − T + 6 X − T + 2 X − T − X − T − X − T ) / ((1 − X − T ) (1 − T )(1 − XT )) . A routine calculation shows that, even though P is not a cograph, W − P ∨ P ( X, T ) isindeed correctly calculated by (8.2) for Γ = Γ = P .While we defined cographs in terms of disjoint unions and joins, either one of thesetwo operations could be replaced by complements. For instance, the class of cographs isthe smallest class of graphs that contains a single vertex and which is closed under takingdisjoint unions and complements. Write ˆΓ for the complement of a simple graph Γ. Question 10.2 (Complements) . Is there an involution W ( X, T ) ˆ W ( X, T ) of rationalgenerating functions in T such that W − ˆΓ ( X, T ) = ˆ W − Γ ( X, T ) for each simple graph Γ? We noted in §5.2 that the class of functions W H ( X, T ) attached to hypergraphs is closedunder Hadamard products. However, it remains an open problem to exploit the Hadamardfactorisation (5.15) in a way that improves upon the general Theorem C. For disjointunions BH n , m of block hypergraphs, we achieved this in (5.18). Even in the special case n = m , the latter formula seems to admit substantial further improvements. We illustratethis for n = m = ( a, . . . , a ). Recall from [57, §5.4] the definitions of the statistics N andd B on the group B r = {± } o S r of signed permutations of degree r .104 Proposition 10.3. Let a = ( a, . . . , a ) ∈ N r . Then W BH a , a ( X, T ) = P σ ∈ B r ( − X − a ) N( σ ) T dB( σ ) (1 − T ) r +1 . (10.1) Proof. The proof of [57, Corollary 5.17] of the case a = 1 (a consequence of a result dueto Brenti [12, Theorem 3.4]) carries over to a general a ; just replace − q − by − q − a . (cid:7) The intriguing shape of the numerator of the right-hand side of (10.1) prompts thefollowing. Question 10.4. Is there an interpretation of the rational functions(a) W BH n , n ( X, T ) in (5.18) for n ∈ N r ,(b) W BH n , m ( X, T ) in (5.18) for n , m ∈ N r , or possibly even(c) W H ( µ ) ( X, T ) in (5.12) for general µ ∈ N P ( V )0 in terms of statistics on Weyl (or more general reflection) groups?Note that (b) includes, as a special case, the problem of finding an interpretation ofthe class counting zeta functions ζ cc G K n ⊗ O ( s ) in terms of permutation statistics; see §8.2.1. The determination of the rational functions W − Γ ( X, T ) for all simple graphs on at mostseven vertices inspired us to raise the following. Question 10.5 (Characterising kites) . Let Γ be a simple graph. Are the followingproperties equivalent?(a) Γ is a kite graph.(b) W − Γ ( X, T ) is a product of factors of the form (1 − X A T B ) ± .The implication (a) → (b) in Question 10.5 follows from Theorem 8.19. Let γ − bethe negative adjacency representation of Γ. Then (b) is equivalent to ζ ask γ − ( s ) factoringas a product of factors ζ ( Bs − A ) ± , where ζ denotes the Riemann zeta function. Inparticular, class counting zeta functions of cographical group schemes associated withkite graphs over rings of integers of number fields admit meromorphic continuation to C .One may speculate that the condition in Question 10.5(b) is not just sufficient butalso necessary for the Euler product ζ cc G Γ ⊗O K ( s ) = Y v ∈V K W − Γ ( q v , q | E(Γ) |− sv )associated with an arbitrary simple graph Γ and number field K with ring of integers O K to admit meromorphic continuation to the whole complex plane. Indeed, consider an105 EFERENCES Euler product Q v ∈V K f ( q v , q − sv ), where f ( X, T ) ∈ Z [ X, T ] is a fixed polynomial. Then ageneral conjecture based on work of Estermann [28] and Kurokawa [40, 41] predicts thatsuch an Euler product admits meromorphic continuation to all of C if and only if f ( X, T )is a product of unitary polynomials; cf. [22, Conjecture 1.11] for details and related work.In a similar spirit, recall from Theorem 5.26 that for a hypergraph H with incidencerepresentation η , we denoted the common abscissa of convergence of ζ ask η O K ( s ) for eachnumber field K by α ( H ). By [57, Theorem 4.20], there exists a positive real number δ ( H ),independent of K , such that the function ζ ask η O K ( s ) can be meromorphically continued tothe domain { s ∈ C : Re( s ) > α ( H ) − δ ( H ) } . Question 10.6. What is the largest value of δ ( H ) (if such a value exists)? Whendoes ζ ask η O K ( s ) admit meromorphic continuation to all of C for each number field K ?By Proposition 5.9, staircase hypergraphs have the latter property. Acknowledgements The work described here was begun while we both enjoyed the generous hospitality ofour mutual host Eamonn O’Brien and the University of Auckland. 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Wilf, generatingfunctionology , Third edition, A K Peters, Ltd., Wellesley, MA, 2006.Tobias RossmannSchool of Mathematics, Statistics and Applied MathematicsNational University of Ireland, GalwayGalwayIrelandE-mail: [email protected] Christopher VollFakultät für MathematikUniversität BielefeldD-33501 BielefeldGermanyE-mail: [email protected], Third edition, A K Peters, Ltd., Wellesley, MA, 2006.Tobias RossmannSchool of Mathematics, Statistics and Applied MathematicsNational University of Ireland, GalwayGalwayIrelandE-mail: [email protected] Christopher VollFakultät für MathematikUniversität BielefeldD-33501 BielefeldGermanyE-mail: [email protected]