Some evidence for the Coleman-Oort conjecture
aa r X i v : . [ m a t h . AG ] F e b SOME EVIDENCE FOR THE COLEMAN-OORTCONJECTURE
DIEGO CONTI, ALESSANDRO GHIGI, ROBERTO PIGNATELLI
Abstract.
The Coleman-Oort conjecture says that for large g there areno positive-dimensional Shimura subvarieties of A g generically containedin the Jacobian locus. Counterexamples are known for g ≤
7. They canall be constructed using families of Galois coverings of curves satisfyinga numerical condition. These families are already classified in caseswhere: a) the Galois group is cyclic, b) it is abelian and the family is 1-dimensional, and c) g ≤
9. By means of carefully designed computationsand theoretical arguments excluding a large number of cases we are ableto prove that for g ≤
100 there are no other families than those alreadyknown. Introduction A g the moduli space of principally polarized complex abelianvarieties of dimension g , by M g the moduli space of smooth complex alge-braic curves of genus g and by j : M g → A g the period mapping (or Torellimapping), which associated to [ C ] ∈ M g the moduli point of the Jacobianvariety J C provided with the theta polarization. The
Jacobian locus is theimage j ( M g ). By j ( M g ) we denote the closure of j ( M g ) in A g .On A g there is a tautological Q -variation of the Hodge structure (in theorbifold sense): if A is a principally polarized abelian variety, the fibre overits moduli point [ A ] ∈ A g is H ( A, Q ) with its Hodge structure of weight 1.In general, given a variation of the Hodge structure H → B , it is interestingto consider the points b ∈ B where the Hodge structure is “more symmetric”than over the general point. Making precise the meaning of “more symmet-ric” requires some effort. In the simplest case this means that the Hodgestructure has more automorphism than usual. For example for the variationover A , the general point has no automorphisms beyond {± } , while thepoints with more automorphisms represent the well-known elliptic curveswith automorphisms Z / Z or Z / Z . The general case is more complicatedsince the symmetry is not at the level of automorphisms but is detected byHodge classes in general tensor spaces. The loci obtained in this way are Mathematics Subject Classification.
Primary: 14G35, 14J10, 14Q05, Secondary:20F99,The authors were partially supported by INdAM (GNSAGA). The second author waspartially supported also by MIUR PRIN 2015 “Moduli spaces and Lie Theory” , by MIURFFABR, by FAR 2016 (Pavia) “Variet`a algebriche, calcolo algebrico, grafi orientati etopologici”, by MIUR, Programma Dipartimenti di Eccellenza (2018-2022) - Dipartimentodi Matematica “F. Casorati”, Universit`a degli Studi di Pavia. The third author waspartially supported also by MIUR PRIN 2015 “Geometry of Algebraic Varieties” and byMIUR PRIN 2017 “Moduli Theory and Birational Classification”. called the
Hodge loci of the variation of the Hodge structure. In the caseof A g they are also called special subvarieties or Shimura subvarieties . (See[26, § Z ⊂ A g is said to be generically contained in j ( M g ) if Z ⊂ j ( M g ) and Z ∩ j ( M g ) = ∅ . Arithmetical considerations ledfirst Coleman and later Oort [27] to the following Conjecture 1.2 (Coleman-Oort) . For large g there are no special subvari-eties of positive dimension generically contained in j ( M g ) . (See [26, §
4] for more details.) This expectation is also motivated by an-other stronger expectation originating from the point of view of differentialgeometry: special subvarieties are totally geodesic with respect to the locallysymmetric (orbifold) metric on A g (the one coming from the Siegel space).If one believes that j ( M g ) bears no strong relation to the ambient geometryof A g , in particular that it is very curved inside A g , then it is natural toexpect that j ( M g ) contains generically no totally geodesic subvarieties, andin particular no Shimura subvarieties (see [8], [17], [15] for results in thisdirection).What makes the problem more interesting is that for low genus examplesof such Shimura varieties generically contained in j ( M g ) do exist! All theexamples known so far are in genus g ≤ First construction . Let G be a finite group acting on a curve C .Consider the family of curves C → B with a G -action of the same topologicaltype (see below for the precise definition). For every m , H ( C b , mK C b ) isa representation of G and its equivalence class is independent of b ∈ B .Denote by B ′ ⊂ M g the moduli image of B and by Z the closure of j ( B ′ ) in A g . In [12, 13] it is proven that ifdim( S ( H ( K C b ))) G = dim H (2 K C b ) G , ( ∗ )then Z is a Shimura variety generically contained in j ( M g ). We also saythat the family of G -covers C → B yields a Shimura variety to mean that Z is Shimura. We refer to such a Shimura variety as a counter-example toColeman-Oort conjecture. Several counter-examples are known, see Theo-rem 1.5 below.1.4. Second construction . Consider a Shimura variety Z generically con-tained in j ( M g ) obtained as in 1.3 from a family of G -curves C → B . Denoteby g ′ the genus of C b /G . Let Nm : J C b → J ( C b /G ) be the norm map ofthe covering f b : C b → C b /G , defined by Nm( P i p i ) := P i f b ( p i ), Then(ker Nm) ⊂ J C b is an abelian subvariety, the generalized Prym variety ofthe covering f b . The theta polarization of J C b restricts to a polarization ofsome type δ on the Prym variety. We get maps ϕ : B −→ M g , ϕ ( b ) := [ C b /G ] , P : B −→ A δg − g ′ , P ( b ) := [(ker Nm) ] . P is the generalized Prym map . If g ′ = 0 the map ϕ is of course constant, A δg − g ′ = A g and P is just the Torelli map, so we get nothing new. If instead OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 3 g ′ >
0, the irreducible components of the fibres of P and ϕ are totallygeodesic subvarieties and countably many of them are in fact Shimura, see[20] and [14, Thm. 3.9, Thm. 3.11]. Thus for g ′ > j ( M g ) and countably many Shimura varieties generically con-tained in j ( M g ).Let us summarize what is known about the counter-examples obtainedvia these constructions. Theorem 1.5. a) There are 38 families of Galois coverings of the pro-jective line satisfying ( ∗ ) with ≤ g ≤ . For g ≤ there are noother counter-examples. See [30, 25, 26, 12] .b) There are 6 families of Galois coverings of elliptic curves satisfying ( ∗ ) with ≤ g ≤ . For g ≤ there are no other counter-examples.See [13] .c) If a family satisfies ( ∗ ) and g ′ > , then necessarily g ′ = 1 and thefamily is one of those in (b). See [14] . g ≥
2, since for g = 1 there are infinitely many1-dimensional families satisfying ( ∗ ).In fact, for every elliptic curve C the involution p
7→ − p acts triviallyon both S H ( K C ) and H (2 K C ). Let G be the group of the biholomor-phisms of C generated by it and by a finite group of translations. Then S H ( K C ) G = S H ( K C ) ∼ = C ∼ = H (2 K C ) = H (2 K C ) G , so giving exam-ples of ( ∗ ) with G of order arbitrarily high. Two of these families are listedin Table 2 in [12].However all these families are irrelevant for the Coleman-Oort conjecture,since in all cases B ′ = M . Note also that some of the families of Theorem1.5 yield the same Shimura variety, i.e. have the same image in moduli, see[12, 13].1.7. It follows from Theorem 1.5 (c) that all the cases where ( ∗ ) holds and g ′ > g ′ = 0, i.e. C b /G = P .The purpose of this paper is to provide further evidence for the Coleman-Oort conjecture, employing a computational approach complemented by the-oretical arguments. Our result is the following improvement of Theorem 1.5. Theorem 1.8.
The positive-dimensional families of Galois covers satisfying ( ∗ ) with ≤ g ≤ are only those of Theorem 1.5. no new families at all is strong evidence thatthere are no more families satisfying ( ∗ ). Since all known counter-examplesto the Coleman-Oort conjecture can be constructed using these families, thisalso suggests that either further counter-examples do not exist or they areof a completely different nature. DIEGO CONTI, ALESSANDRO GHIGI, ROBERTO PIGNATELLI G -covers are identified by data of combinatorial and group-theoretical nature. We explain this in §
2. So the basic strategy is obviouslyto list all these data and check condition ( ∗ ) for each datum in the list.Since the list of these data is extremely long, one needs to avoid unnecessarycomputations. The first observation is that many data give rise to the samefamily. More precisely call two data ∆ and ∆ ′ Hurwitz equivalent if they havethe same group G and if the families corresponding to them are isomorphicas families of algebraic curves with G -action. It turns out that Hurwitzequivalence classes can be huge. To check condition ( ∗ ) for all the families ofsome genus, one would start by choosing a representative out of any Hurwitzequivalence class, and proceed by checking ( ∗ ) for all the representatives.However, the identification of a single representative inside each class is adaunting task, since the classes are huge and Hurwitz equivalence is rathercomplicated. (An algorithm dealing with Hurwitz equivalence appears in [2].It was used in [12] and [13]. An improvement of this algorithm is given in[3]. We hope to address the problem of algorithmic computation of Hurwitzequivalence in future work.)Luckily there is another equivalence relation on data, much coarser thanthe Hurwitz equivalence, which is appropriate to our problem: if ∆ =( G, g , . . . , g r ), then the number N = N (∆) only depends on the conju-gacy classes C = [ g ] , . . . , C r = [ g r ]. Also the order of these is completelyirrelevant. The unordered sequence ( C , . . . , C r ) is called a refined passport .(See Definition 3.7.) So our problem depends only on refined passports, moreprecisely on their Aut( G )-orbits, which are considerably less in number thanHurwitz equivalence classes, leading to much shorter execution times. No-tice that in some cases refined passports (even if taken up to the action ofAut( G )) are still too many to be stored simultaneously into memory, butthis is not a problem, since we only need to perform an iteration to check( ∗ ) on each individually.Even after this great simplification the computation remains quite formi-dable, at least for the computers at our disposal. We use a number of tricksto reduce the data that must be considered. Several exclusions (e.g. cyclicgroups) follow from previous results (see Theorem 3.3). We complementthem with Corollary 3.13, which effectively eliminates more than 90% of thedata, including some of the hardest cases, thus allowing us to complete thecomputation.1.11. For the implementation of the algorithm we used MAGMA [23], which isquite suited to the task at hand since it allows working with groups, groupactions and representations, in particular computing characters, orbits andstabilizers; furthermore, it contains a database of groups of small order. Ourcode is available at [9].The problem lends itself easily to parallelization, since each group andsignature is treated independently; however,
MAGMA does not support par-allelization natively. The first part of the computation (Algorithm 1) wasparallelized using the standard tool [32]. On the other hand, the rest ofthe computation can become quite memory-intensive; this leads to technicaldifficulties, mainly concerning situations in which one of the processes is
OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 5 terminated for lack of memory, which were addressed by writing an ad hocexternal program to run the
MAGMA script.Using a computer with 56 Intel Xeon 2.60GHz CPU and 128 GB of RAMwe were able to finish the computations in less than three days.1.12. An important point to stress is the following. Condition ( ∗ ) is sufficientfor a family to yield a Shimura variety. In general it is unknown if it is alsonecessary. In this paper we only check whether condition ( ∗ ) holds. So wecannot exclude that these families give rise to counter-examples to Coleman-Oort conjecture.1.13. The plan of the paper is as follows. In § G -curves and some basic facts concerning the multiplicationmap on sections of the canonical bundle, which is related with condition( ∗ ). At the end we prove Lemma 2.17, which deals with the behaviour ofcondition ( ∗ ) when passing from a given family to a quotient by a normalsubgroup. In § § Acknowledgements.
The authors would like to thank Paola Frediani forhelp with Lemma 2.17 and Matteo Penegini and Fabio Perroni for interestingdiscussions related to the subject of this work. The second author would liketo thank Matteo Garofano and Gabriele Merli for technical help with theinstallation and the maintenance of the server used for the computations.2.
Families of G -curves C ∆ → B ∆ the corresponding family of curves. The image of B ∆ in M g will be denotedby M ∆ . We are interested in the closure of M ∆ in A g . As explained in 1.3,when ( ∗ ) holds this closure is a Shimura variety generically contained in theJacobian locus. This is explained in more detail at the end of this section,together with some related remarks on the multiplication map.In the following, unless otherwise stated, we assume that the genus is atleast 2. For r ≥
3, set Γ r := h γ , . . . , γ r | r Y i =1 γ i = 1 i . Definition 2.2. If G is a finite group an epimorphism θ : Γ r → G is called admissible if θ ( γ i ) = 1 for i = 1 , . . . , r . An r -datum is a pair ∆ = ( G, θ ) where G ∈ G and θ : Γ r → G is an admissible epimorphism. The signature of ∆ is the vector m := ( m , . . . , m r ) where m i := ord( θ ( γ i )) . The genus of ∆ is defined by the Riemann-Hurwitz formula: g (∆) −
1) = | G | − r X i =1 (cid:18) − m i (cid:19)! (2.1) DIEGO CONTI, ALESSANDRO GHIGI, ROBERTO PIGNATELLI
We let D r or simply D denote the set of all r -data. S by the outer normal. Consider smooth regular arcs ˜ α i in S joining p to p such that for i = j ˜ α i and ˜ α j intersect only at p . Assumealso that the tangent vectors at p are all distinct and follow each other incounterclockwise order. Next consider loops α i based at p constructed asfollows: α i starts at p , travels along ˜ α i until near p i , there travels coun-terclockwise along a small circle around p i , finally goes back to p i againalong ˜ α i . The circles have to be pairwise disjoint. We call the resulting setof generators { [ α ] , . . . , [ α r ] } a geometric basis of π ( S − P, p ). Once ageometric basis is fixed, there is a well-defined isomorphism χ : Γ r → π ( S − P, p )such that χ ( γ i ) = [ α i ].2.4. The following geometric setting gives rise to data (and it is the mainmotivation for them). Let X be a compact (connected) Riemann surface.Assume that a finite group G acts effectively and holomorphically on X insuch a way that X/G = P . Let P := { p , . . . , p r } be the critical values of π : X → P ∼ = S . Fix p ∈ S − P and a geometric basis { [ α ] , . . . , [ α r ] } with corresponding isomorphism χ : Γ r ∼ = π ( S − P, p ). Finally fix a point˜ p ∈ π − ( p ). As is well-known there is a morphism ˜ θ : π ( S − P, p ) → G such that for [ α ] ∈ π ( S − P, p ) the lifting of α starting at p ends at g · p where g = ¯ θ ([ α ]). Since X is connected ˜ θ is surjective. Therefore∆ := ( G, θ := ˜ θ ◦ χ ) is an r -datum, g (∆) = g ( X ) by the Riemann-Hurwitzformula and m i is the cardinality of the stabilizer of points in π − ( p i ). Weare going to show that each datum arises from a covering X → P = X/G .2.5. Assume from now on that r ≥ T ,r the Teichm¨ullerspace in genus 0 and with r marked points. The definition of T ,r is asfollows. Fix r + 1 distinct points p , . . . , p r on S . For simplicity set P =( p , . . . , p r ). Consider triples of the form ( P , x, [ f ]) where x = ( x , . . . , x r )is an r -tuple of distinct points in P and [ f ] is an isotopy class of ori-entation preserving homeomorphisms f : ( P , x ) → ( S , P ). Two suchtriples ( P , x, [ f ]) and ( P , x ′ , [ f ′ ]) are equivalent if there is a biholomor-phism ϕ : P → P such that ϕ ( x i ) = x ′ i for any i and [ f ] = [ f ′ ◦ ϕ ]. TheTeichm¨uller space T ,r is the set of all equivalence classes, see e.g. [1, Chap.15] for more details.2.6. Fix a geometric basis B = { [ α i ] } ri = of π ( S − P, p ) with correspondingisomorphism χ : Γ r ∼ = π ( S − P, p ). Given an r -datum ∆ = ( G, θ ), theepimorphism θ ◦ χ − gives rise to a topological covering π : Σ → S − P . Bythe topological part of Riemann’s Existence Theorem this can be completedto a branched cover π : Σ → S . Given a point t = [ P , x, [ f ]] ∈ T ,r , thehomeomorphism f restricts to a homeomorphism of P − x onto S − P .We get an induced isomorphism f ∗ : π ( P − x, f − ( p )) ∼ = π ( S − P, p ).Thus θ ◦ χ − ◦ f ∗ : π ( P − x, f − ( p )) → G is an epimorphism and thisgives rise to a topological covering π t : C t → P − x . Here C t is an open OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 7 differentiable surface. Since π is a local diffeomorphism, there is a uniquecomplex structure on C t making π t holomorphic. By the holomorphic partof Riemann’s Existence Theorem C t and π t may be uniquely completed toa proper holomorphic map π t : C t → P and the G -action extends to C t .Moreover there is an isotopy class of homeomorphisms ˜ f t : C t → Σ thatcover f t . As t varies in T ,r this construction yields a holomorphic map tothe Teichm¨uller space of ΣΦ ∆ : T ,r −→ T g ∼ = T (Σ) , t [ C t , [ ˜ f t ]] . The group G embeds in the mapping class group of Σ, which we denoteby Mod g . This embedding depends on θ and we denote by G θ ⊂ Mod g itsimage. The image of Φ ∆ coincides with T G θ g , the set of fixed points of G θ on T g . As such it is a complex submanifold. We denote it by T ∆ .The image of T ∆ in the moduli space M g is an irreducible algebraic sub-variety of dimension ( r −
3) that we denote by M ∆ . (See e.g. [19, 6, 2, 7]for more details.) As explained in [19, p. 79] the map T ∆ → M ∆ factorsthrough an intermediate variety ˜ M ∆ : T ∆ −→ ˜ M ∆ ν −→ M ∆ . The variety ˜ M ∆ is the normalization of M ∆ . There is a finite cover B ∆ → ˜ M ∆ and a universal family π ∆ : C ∆ → B ∆ . We call it the family of G -curves associated to ∆. The proofs of theseassertions can be found in [19] (where T g ( H ) corresponds in our notationto T ∆ , f M ( H ) to ˜ M ∆ , M ( H ) to M ∆ and ˜ M pure ( H ) to B ∆ ). Note thatdim M ∆ = dim B ∆ = r − . (2.2)2.7. In this construction the choice of the base point p is irrelevant. Infact (up to isomorphism) the ramified covering Σ → S only depends on N := ker θ ◦ χ − ⊳ π ( S − P, p ). Two isomorphism π ( S − P, p ) → π ( S − P, p ′ ) differ by an inner automorphism, so the map from normalsubgroups of π ( S − P, p ) to those of π ( S − P, p ′ ) is well defined. Thisproves that T ∆ and hence also M ∆ , ˜ M ∆ and the family π ∆ : C ∆ → B ∆ donot depend on the choice of the base point p .2.8. On the other hand the construction of T ∆ , M ∆ , ˜ M ∆ , π ∆ does depend onthe choice of the geometric basis. Let B = { [ ¯ α i ] } ri =1 be another geometricbasis. and let ¯ χ : Γ r → π ( S − P, p ) be the corresponding isomorphism.Then µ := ¯ χ ◦ χ − ∈ Aut π ( S − P, p ) has two special properties: 1) forevery i = 1 , . . . , r , µ ([ α i ]) = [ ¯ α i ] is conjugate to [ α j ] for some j ; 2) theinduced homomorphism on the cohomology group H ( π ( S − P, p ) , Z ) isthe identity. By a variant of the Dehn-Nielsen Theorem (see e.g. [10, § ϕ : ( S − P, p ) → ( S − P, p ) such that µ = ϕ ∗ . Let Σ and¯Σ be the coverings of S obtained from χ and ¯ χ . If N = ker θ ◦ χ − and¯ N = ker θ ◦ ( ¯ χ ) − , then ϕ ∗ ( N ) = ¯ N . By the Lifting Theorem there is anorientation-preserving diffeomorphism ˜ ϕ : Σ → ¯Σ that covers ϕ . This gives DIEGO CONTI, ALESSANDRO GHIGI, ROBERTO PIGNATELLI rise to a biholomorphism T (Σ) → T (Σ ′ ) which maps T ∆ constructed using χ to T ∆ constructed using ¯ χ . The identification T g = T (Σ) is defined upto the action of Mod g and the discussion above shows that also T ∆ is welldefined up to this action. In particular M ∆ , ˜ M ∆ , B ∆ and π ∆ are completelyindependent of the choice of the geometric basis.2.9. There is a representation ρ : G −→ GL H ( C t , K C t ) , ρ ( g ) := ( g − ) ∗ . (2.3)The equivalence class of this representation is independent of t ∈ B ∆ .For later use we recall the following observation, already used in the proofof [14, Thm. 2.3]. Proposition 2.10.
Let G be a finite group of automorphisms of a curve C ,and consider the subspace of invariants H ( C, K C ) G . Then the multiplica-tion map m GC : S H ( C, K C ) G → H ( C, K C ) G is surjective unless C is hyperelliptic (so of genus at least ) and there is asmall deformation C t of the complex structure of C such that all elementsof G remain holomorphic and the general curve C t is not hyperelliptic.In particular, for a fixed r -datum ∆ = ( G, θ ) , the map m GC is surjectivefor the general C ∈ B ∆ .Proof. Let g be the genus of C . The statement is obvious for g ≤ G − equivariant map S ( H ( C, K C )) → H ( C, K C ) is an isomorphism(among spaces of dimension g ). If C is not hyperelliptic, then the statementfollows similarly since the map S ( H ( C, K C )) → H ( C, K C ) is surjectiveby M. Noether’s Theorem.We can then assume that C is hyperelliptic. Let σ be the hyperellipticinvolution. It is well-known that σ acts as the multiplication by − H ( C, K C ), so trivially on S ( H ( C, K C )), and that the multiplication map S ( H ( C, K C )) → H ( C, K C ) h σ i is surjective.We distinguish two cases.(1) If σ ∈ G then the surjectivity of m GC follows by the surjectivity ofthe map S ( H ( C, K C )) → H ( C, K C ) h σ i .(2) If σ G we denote by ˜ G the group of automorphisms of G generatedby G and σ . Then m ˜ GC is surjective. Moreover S ( H ( C, K C )) ˜ G = S ( H ( C, K C )) G so we need H ( C, K C ) G ∼ = H ( C, K C ) ˜ G , thatis equivalent to H ( C, K C ) G ⊂ H ( C, K C ) h σ i . Dualizing, this isequivalent to H ( C, T C ) G ⊂ H ( C, T C ) h σ i , which amounts to askingthat every small deformation of the pair ( C, G ) remain hyperelliptic. (cid:3) Z / Z ) intersects the hyperelliptic locus in the 2-dimensional family of curves withan action of ( Z / Z ) considered in [29, Table 2 - Five critical values - (b)]. OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 9 If C belongs to this latter family, then 3 = h ( C, K C ) G = h ( C, K C ) ˜ G = 2and therefore m GC has corank 1.2.12. Consider now a datum ∆ and the family π ∆ : C ∆ → B ∆ . As t variesin B ∆ , the domain and codomain of m GC t do not change in dimension. Set N (∆) := dim (cid:0) S H ( C t , K C t ) (cid:1) G (2.4) Theorem 2.13. If g = g (∆) ≥ and N (∆) = r − , ( ∗ ) then j ( M ∆ ) (closure in A g ) is a special subvariety of PEL type of A g that isgenerically contained in the Jacobian locus. (See [12, Thm. 3.9] and [13, Thm. 3.7].)2.14. The idea of Theorem 2.13 is that from ∆ one can construct both M ∆ and a Shimura subvariety Z ∆ ⊂ A g with N (∆) = dim Z ∆ . By construction j ( M ∆ ) ⊂ Z ∆ and both M ∆ and Z ∆ are irreducible algebraic subvarieties. By(2.2) dim M ∆ = r −
3. Since j is an injective morphism of algebraic varieties,when g ≥ N ≥ r −
3. If ( ∗ ) holds, then j ( M ∆ ) is dense in Z ∆ .2.15. Note also that (when g ≥
2) for any t ∈ B ∆ we have dim H (2 K C t ) G =dim H ( T C t ) G = dim B ∆ = r −
3. Hence condition ( ∗ ) in Theorem 2.13coincides with condition ( ∗ ) of the Introduction. It amounts to asking thatdomain and codomain of m GC t have the same dimension. By Proposition 2.10this is then equivalent to asking that, for general t , m GC t is injective.2.16. We now wish to prove a lemma that is helpful to rule out a priori some groups.Let ∆ = ( G, θ ) be a datum and let H be a normal subgroup of G . Set K := G/H and let π : G → K be the canonical projection. The composition π ◦ θ : Γ r → G → K is an epimorphism, but it is not necessary admissible,since some of the γ i ∈ Γ r might map to 1. We can throw them awayobtaining an admissible epimorphism ¯ θ : Γ s → K for some s ≤ r . In termsof spherical generators this means the following: if θ ( γ i ) = g i and k i = π ( g i ),then ¯ θ = ( k , . . . , k r ) where we omit all the k i that equal 1. So we get a newdatum ¯∆ = ( K, ¯ θ ). This corresponds to the following geometric situation.∆ gives rise to the family π ∆ : C ∆ → B ∆ . We can quotient each fibre C t by H getting a curve F t := C t /H on which K acts: C t F t = C t /H P = C t /G = F t /K. π p The curves F t form a family F → B ∆ . If g ( F t ) ≥
2, out of the datum ¯∆ wecan form the family C ¯∆ → B ¯∆ as explained in 2.6. Then F is a pull-backof this family, i.e. f ∗ C ¯∆ = F for some holomorphic map f : B ∆ → B ¯∆ . Lemma 2.17.
In the above situation, assume that g ( F ) ≥ . If ( ∗ ) holdsfor ∆ , then it holds also for ¯∆ .Proof. Write for simplicity C = C t and F = F t . We have two pull-backmaps: p ∗ : H ( K F ) ֒ → H ( K C ) , p ∗ : H (2 K F ) ֒ → H (2 K C ) . From the first one we obtain also an injection f := S p ∗ : S H ( K F ) ֒ → S H ( K C ) . Since p ∗ H ( K F ) = H ( K C ) H then f (( S H ( K F )) K ) ⊂ ( S H ( K C )) G . Thus, we get a commutative diagram (cid:0) S H ( K F ) (cid:1) K H (2 K F ) K (cid:0) S H ( K C ) (cid:1) G H (2 K C ) G . f m KF p ∗ m GC from which m GC injective ⇒ m KF injective . As explained in 2.15, if ( ∗ ) holds for ∆, then m GC is injective for general C and therefore m KF is injective for general F , so N ( ¯∆) = S H ( K F ) K ≤ H (2 K F ) K . But since g ( F ) ≥
2, the discussion in 2.14 shows that N ( ¯∆ ≥ s − H (2 K F ) K . Thus N ( ¯∆) = s −
3, i.e. ¯∆ satisfies ( ∗ ). (cid:3) Avoiding unnecessary computations
This section collects several results that allow to rule out a priori variouscases avoiding some parts, sometimes really substantial, of the computation.We briefly explain its contents.Lemmata 3.1 and 3.2 use the same ideas underlying the proof of the Hur-witz theorem to ensure that signatures exist only in some ranges. Theorem3.3) summarizes results of Moonen and Mohajer-Zuo, saying that no newcounter-examples exist in certain cases.In 3.4 we introduce spherical systems of generators, recall the Chevalley-Weil formula, define refined passports and show that N (∆) only depends onthe refined passport of the generators. We then recall Eichler’s formula. Itis used in the proof of Theorem 3.12, which says that no counter-exampleexists with G = ( Z / Z ) k for g ≥
4. Its Corollary 3.13 is the main toolto cut down the number of computations to be done. Other such toolsare Frobenius’ test (Corollary 3.15) and an elementary observation on theabelianization of a group admitting a spherical system of generators ( § Lemma 3.1. If ( G, θ ) is an r -datum of genus g and G contains an elementof order > g − , then either r = 3 , i.e. the family is -dimensional, orit coincides with family (5) in [12, Table 2] . OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 11 If x ∈ G has order > g − H := h x i is a large auto-morphism group of C . So the Lemma follows immediately from Proposition4.5 in [21]. The idea of using upper bounds for the order of single elementsof G comes from Corollary 5.10 in [4], where the classical bound of Wimanwas used. The theorem of Kulkarni that we use here is more precise. Lemma 3.2.
Let ∆ = (
G, θ ) be an r -datum with genus g ≥ and r ≥ . Ifthe datum corresponds to an action of G on a smooth curve X with X/G = P , then (a) r ≤ g +2 with equality only for X hyperelliptic and G generatedby the hyperelliptic involution, (b) r ≤ g − d and (c) | G | ≤ g − .Proof. The arguments are extremely classical, but for the reader’s conve-nience we give the proof. Set d := | G | , δ := P ri =1 1 m i and µ := r − − δ . Bythe Riemann-Hurwitz formula,2( g −
1) = d · µ (3.1)Assume 2 ≤ m ≤ m ≤ · · · ≤ m r . Since g ≥ µ >
0. For x > f ( x ) := 1 − /x . Then µ = P ri − f ( m i ) −
2. Since f is increasing µ ≥ r · f (2) − r − /
2. Using d ≥ g − ≥ ( r − / | G | = 2, so the curves arehyperelliptic. By a dimensional count the family coincides with that ofhyperelliptic curves. This proves (a).Set A ( r ) := { x = ( x , . . . , x r ) ∈ Z r : x i ≥ , P ri − f ( x i ) > } and¯ µ ( r ) := min x ∈ A ( r ) (cid:26) r X i =1 f ( x i ) − (cid:27) Using the fact that f is strictly increasing one verifies that for r = 4 theminimum is achieved at x = (2 , , ,
3) and ¯ µ (4) = 1 /
6, while for r ≥ x = (2 , . . . , | {z } r times and ¯ µ ( r ) = r/ −
2. So for any r ≥ µ ( r ) ≥ ( r − /
2. Let now m be the signature of the datum ( G, θ ).Then m ∈ A ( r ), so µ ≥ ¯ µ ( r ). Thus (3.1) gives 2( g − /d ≥ ¯ µ ( r ) ≥ ( r − / r = 4, (3.1) gives 2( g − /d ≥ ¯ µ (4) = 1 /
6, which is equivalent to theinequality in (c). If r > d ≤ g − / ( r − ≤ g − g − / ( r − ≤ g − ≤ g − r . (cid:3) Theorem 3.3.
The data ∆ = (
G, θ ) satisfying ( ∗ ) with G cyclic or with G abelian and r = 4 are Hurwitz equivalent to those mentioned in Theorem 1.5.Moreover for such data ( ∗ ) is necessary for Z ∆ to be a Shimura subvariety. These results are due to Moonen [25] and Mohajer-Zuo [24, Thms. 3.1 and6.2].3.4. If G is a finite group, giving an r -datum ∆ = ( G, θ ) is equivalent togiving a list of generators g , . . . , g r of G such that g i = 1 for any i andsubject to the constraint g · · · g r = 1. Indeed, this defines an epimorphism θ : Γ r → G by θ ( γ i ) = g i . From now on we will write ∆ ∈ D r as ∆ =( G, g , . . . , g r ), and we will call ( g , . . . , g r ) a spherical system of generators of the group G . Let χ ρ denote the character of the representation ρ defined in (2.3). Asexplained in [12, §§ N (∆) in (2.4) can be computed from χ ρ : N (∆) = 12 | G | X a ∈ G (cid:0) χ ρ ( a ) + χ ρ ( a ) (cid:1) . (3.2)So to test ( ∗ ) one needs to compute χ ρ for a datum ∆. There are two waysto do that: using Eichler’s trace formula or the Chevalley-Weil formula. Weneed both and we start from the Chevalley-Weil formula.3.5. Next, fix a datum ∆ = ( G, g , . . . , g r ) and let m j := ord( g j ) as usual.Denote by Irr G the set of irreducible characters of G . For each χ ∈ Irr G fixa representation σ χ with character χ . For n ∈ N , n > ζ n := exp(2 πi/n ).If χ ∈ Irr G , 1 ≤ j ≤ r and 0 ≤ α < m j , denote by N j,α the multiplicity of ζ αm j as an eigenvalue of σ χ ( g j ). Theorem 3.6 (Chevalley–Weil) . If ∆ = ( G, g , . . . , g r ) is a datum for theGalois covering C → P , then the multiplicity µ χ of χ ∈ Irr G in ρ is (3.3) µ χ = − deg χ + r X j =1 m j − X α =0 N j,α αm j + ε, where ε = 1 if χ is the trivial character and ε = 0 otherwise. A nice reference for the Chevalley-Weil formula is [18, Ch. 1]. Our im-plementation uses this formula to compute χ ρ and hence N (∆). In fact weuse the same algorithm as Gleißner, which is based in turn on [31], but withcode optimized for our setting (see 4.6). Definition 3.7.
Given a finite group G let C G or simply C denote the setof conjugacy classes of G . The symmetric group Σ r acts on C rG . A refinedpassport with r branch points for the group G is an element of C rG / Σ r . Thusa refined passport is an undordered sequence of conjugacy classes of G . Given ∆ = ( G, g , . . . , g r ) , the refined passport of ∆ is the class of ([ g ] , . . . , [ g r ]) in C rG / Σ r . Note that this definition is slightly different from those of [22] and [28]: wedo not assume that a refined passport comes from a datum.3.8. It is clear that the numbers N j,α defined in 3.5 do not change if g j is replaced by another element g ′ j ∈ G which is conjugate to g j . Anotherobservation is that obviously the sum in (3.3) is independent of the order.Thus N (∆) depends only on the refined passport of ∆. This elementaryobservation is at the basis of our approach to the computation. Lemma 3.9.
Let G be a finite group and let C i ∈ C G for i = 1 , . . . , r .Assume that there is a datum ∆ = ( G, g , . . . , g r ) with g i ∈ C i for i =1 , . . . , r . Then for any σ ∈ Σ r there is a datum ( G, γ , . . . , γ r ) such that γ i ∈ C σ i for i = 1 , . . . , r . OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 13
Proof.
Since Σ r is generated by simple transpositions, it is enough to provethe result for σ = ( j, j + 1), 1 ≤ j < r . Set γ i = g i , for i
6∈ { j, j + 1 } , γ j = g j g j +1 g − j , γ j +1 = g j . Then (
G, γ , . . . , γ r ) is still a datum and γ i ∈ C σ i for any i . (cid:3) a ∈ G , p ∈ C and a · p = p , then da ( p ) ∈ End T p C is multiplication by a root of unity, which we denote simply by da ( p ). Theorem 3.11 (Eichler Trace Formula) . If a ∈ G , a = 1 then (3.4) χ ρ ( a ) = 1 − X p ∈ Fix( a ) − da ( p ) . See e.g. [11, Thm. V.2.9, p. 264].
Theorem 3.12.
Let ∆ = (
G, g , . . . , g r ) be a datum corresponding to acovering C → P with G ∼ = ( Z / Z ) k . If g ( C ) ≥ , then ( ∗ ) does not hold for ∆ .Proof. The families fulfilling condition ( ∗ ) with genus up to 7 have beenclassified in [12, Theorems 5.4 and 5.5] and are all listed in [12, Table 2]:inspecting the table we see that we may assume g ( C ) ≥ a in G , a = 1, have order 2, by the Hurwitz formula χ ρ (1) = g ( C ) = 1 + | G | r −
4) = 1 + 2 k − ( r − . Moreover for all p ∈ Fix( a ), da ( p ) = − ∈ R and then, by (3.4) for all a ∈ G , χ ρ ( a ) ∈ R . In particular all summands in the expression of N in (3.2) arereal numbers and N (∆) = 12 | G | X a ∈ G (cid:0) χ ρ ( a ) + χ ρ ( a ) (cid:1) = 12 k +1 X a ∈ G (cid:0) χ ρ (1) + χ ρ ( a ) (cid:1) ≥≥ (cid:0)P a ∈ G χ ρ (1) (cid:1) + χ ρ (1) k +1 = g ( C ) (cid:18)
12 + g ( C )2 k +1 (cid:19) == g ( C ) (cid:18)
12 + 12 k +1 + r − (cid:19) = g ( C ) (cid:18) k +1 + r (cid:19) > g ( C ) (cid:16) r (cid:17) ≥ r contradicting ( ∗ ). (cid:3) Considering Lemma 2.17 we deduce the following stronger result:
Corollary 3.13.
Let ∆ = (
G, g , . . . , g r ) be a datum corresponding to acovering C → P . If there is a surjective map G → ( Z / Z ) , then ( ∗ ) doesnot hold for ∆ .Proof. Assume by contradiction that ( ∗ ) holds.Let H be the kernel of the surjection G → ( Z / Z ) and consider thefamily of the curves F t = C t /H → P as in 2.16. They are Galois coverswith datum ¯∆ = (( Z / Z ) , h , · · · , h s ). Since each set of generators of ( Z / Z ) has cardinality at least 4, then s ≥
5. This implies g ( F ) ≥ g ( F ) ≤ P with genus among 2 and 4 having 4 or morebranch points are listed in [12, Table 2]: we see that the group ( Z / Z ) doesnot occur, reaching an absurd. (cid:3) The Galois group G of family (34) in [12, Table 2] admits ( Z / Z ) asa quotient. Thus one cannot improve the above Corollary by substituting( Z / Z ) with one of its proper quotients. In fact applying Lemma 2.17 tothis case yields one of the families of elliptic curves mentioned after Theorem1.5.There is another useful criterion, already used by Breuer [5] and Paulhus[28]. Indeed, for some elements c , one can ascertain a priori that π − ( c ) = p − (˜ c ) does not contain any system of generators at all. This is based on atheorem of Frobenius. (See [22, p. 406] for a proof.) Theorem 3.14 (Frobenius’ formula) . Given a finite group G and conjugacyclasses C , . . . , C r , the number of r -ples ( g , . . . , g r ) ∈ C × · · · × C r suchthat Q g i = 1 is | C | · · · | C r || G | X χ ∈ Irr G χ ( C ) · · · χ ( C r ) χ (1) r − . Notice that this condition is independent of the order.
Corollary 3.15.
Let G be a group and ( C . . . , C r ) a refined passport. If X χ ∈ Irr G χ ( C ) · · · χ ( C r ) χ (1) r − = 0 , then there is no datum ( G, g , . . . , g r ) with refined passport ( C , . . . , C r ) . G admits a system of spherical generators ( g , . . . , g r ) with signature( m , . . . , m r ). Decompose its abelianization Ab G = Z /k Z ⊕ · · · ⊕ Z /k p Z with k | · · · | k p (i.e. the k i ’s are the invariant factors). Since for any j , Ab G is generated by the images of g , . . . , ˆ g j , . . . , g r , it follows that p ≤ r − k p divides lcm( m , . . . , ˆ m j , . . . , m r ) for any j .4. The algorithm G , let C G be the set of its conjugacy classes. Recallfrom Definition 3.7 that a refined passport on G with r branch points is anunordered sequence of r conjugacy classes of G , i.e. an element of C rG / Σ r . Ifa refined passport contains a spherical system of generators ∆ = ( g , . . . , g r ), g (∆) and N (∆) only depend on the refined passport of ∆. We will say thata refined passport is a counter-example of genus g if it contains a sphericalsystem of generators ∆ with g (∆) = g such that ( ∗ ) holds. Notice thatrefined passports that satisfy ( ∗ ) formally but do not contain a sphericalsystem of generators are excluded by this definition. The group Aut G actsboth on C G and on the set of refined passports. OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 15
Problem 4.1.
For fixed g ≥ , list groups G and counter-examples of genus g on G with r ≥ branch points, one for each orbit of Aut( G ) , leaving asidethose with G cyclic and those with G abelian and r = 4 . Our basic strategy is to fix r , and then choose one refined passport of genus g with r branch points in each Aut( G )-orbit. If ( ∗ ) holds, it then sufficesto determine whether the refined passport contains a system of sphericalsystems of generators.4.3. As in [2, 12], we use signature as an invariant. Using the notation ofDefinition 2.2 signature defines a map D r → N r , ( g , . . . , g r ) (ord( g ) , . . . , ord( g r )) . Since the order of an element only depends on its conjugacy class, thesignature of a spherical system of generators ( g , . . . , g r ) only depends onthe conjugacy classes ([ g ] , . . . , [ g r ]). Corresponding to the fact that re-fined passports are taken up to reordering (Lemma 3.9), signatures can beconsidered up to permutation, i.e. we can restrict to signatures satisfying m ≤ · · · ≤ m r .We iterate over the order d = | G | . For fixed d , let S d,g be the set of finitesequences m = ( m , . . . , m r ) such that(S1) 4 ≤ r ≤ g − d + 4 and d ≤ g − m i is a divisor of d ;(S3) 1 < m i < d ;(S4) g and m satisfy (2.1);(S5) m ≤ · · · ≤ m r ;By Lemma 3.2, the signature of a spherical system of generators ∆ with r ≥ g (∆) = g must satisfy (S1); the restriction r ≥ m i < d in (S3)is motivated by the fact that we are only interested in noncyclic groups G .The set of “admissible” signatures S d,g is computed by Algorithm 1. Inthe implementation, we found it convenient to compute each S d,g for 2 ≤ g ≤ g max simultaneously, and then store the result on disk for later retrieval,rather than iterate over g ; this prevents repeating some computations.4.4. Elements of C rG / Σ r (i.e. refined passports) can be viewed as multi-sets . Given a set X , a multiset of elements of X can be defined as a set { ( x , n ) , . . . , ( x k , n k ) } where the x i are pairwise disjoint elements of X andthe n i are nonnegative integers representing the multiplicity of x i . In fact,it is customary to require the n i to be positive, but it will be convenientfor our purposes to allow them to be zero as well. We will write a multisetas { x n , . . . , x n k k } . A set { x , . . . , x k } can be identified with the multiset { x , . . . , x k } , and the union of two multisets is defined in the obvious wayby adding multiplicities.It will also be convenient to represent elements of S d,g as multisets ofintegers { m n , . . . , m n k k } ; for instance, the signature (2 , , , ,
3) will be rep-resented by the multiset { , } . m ∈ S d,g computed in Algorithm 1 and groups G of order d . A refinedpassport with signature m only exists on a group G if there is at least oneelement of order m j for every m j ∈ m ; we therefore discard groups andsignatures that do not satisfy this condition. More groups and signaturescan be eliminated by taking advantage of Lemma 3.1, Corollary 3.13 andthe observation in 3.16. This procedure is displayed in Algorithm 2, whichreduces the problem to identifying counter-examples for fixed group andsignature. Notice that on line 25 the signature { m n , . . . , m n k k } is convertedinto a multiset(4.1) { A n , . . . , A n k k } ⊂ P ( C G ) , where each A i is the subset of C G of conjugacy classes of order m i . This isthe basis for the recursion of Algorithm 4.4.6. At this point we need to determine the counter-examples with a givensignature m and group G . This is achieved by picking one refined passportwith signature m in each Aut( G )-orbit, then verifying whether ( ∗ ) holdsand the refined passport contains a spherical system of generators.The iteration through one refined passport in each Aut( G )-orbit is per-formed in Algorithm 4. A refined passport with signature { m n , . . . , m n k k } isobtained by choosing n i conjugacy classes with order m i for each 1 ≤ i ≤ k ;in terms of (4.1), for each i we must choose a multiset S i of n i elements of A i ,counted with multiplicities. We can write S i in a unique way as a union ofsets S j B ij , where B i ⊃ B i ⊃ . . . is a definitely empty sequence of subsetsof A i ; this means that the multiplicity of C in S i is the number of indices j such that C is in B ij . Thus, iterating through the possible multisets S i isequivalent to iterating through sequences A i ⊃ B i ⊃ B i ⊃ . . . , X | B ij | = n i . This must be repeated for each i = 1 , . . . , k .Our goal is to perform a similar iteration by choosing a single elementin each Aut( G )-orbit. To begin with, our algorithm picks a subset B of A k with 1 ≤ h ≤ n k elements, representing B k in the notation above. Foreach choice of B , the function recursively iterates through refined passportsobtained by taking the union of B and a refined passport with n i elementsin each A i , i < k and n k − h elements in B . The recursive call iteratesthrough one refined passport for each H -orbit, where H is the stabilizer of B in Aut( G ). Top-level iteration over one subset B for each Aut( G )-orbitcompletes the algorithm.This approach requires a much lower amount of memory than determiningall possible refined passports first and then picking one in each Aut( G )-orbit. Notice also that the refined passports produced by the algorithmare elaborated sequentially, and not stored simultaneously into memory.Nevertheless, the algorithm must iterate through one subset of A k for eachAut( G )-orbit, and we are not aware of any efficient way of doing this withoutstoring all subsets of fixed cardinality in memory. This is the one point inthe whole algorithm where memory consumption can be significant. OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 17
Algorithm 3 determines whether a refined passport is a counter-example;first, the condition of Theorem 3.14 is verified, i.e. whether P χ χ ( C ) ··· χ ( C r ) χ (1) r − is nonzero; if so, we will say that ( C , . . . , C r ) passes Frobenius’ test. Then,condition ( ∗ ) is tested by selecting random elements inside each C i andcomputing N (∆) by (3.3). Notice that each term P α N j,α α/m j appearingin (3.3) only depends on the corresponding g j , and the characters χ onlydepend on the group G . Thus, it suffices to compute these data at thebeginning of the computation, when G is fixed, making the computationof (3.3) in the iteration quite fast. Only when both Frobenius’ test and( ∗ ) hold does the algorithm perform the most computationally expensivestep, namely checking whether C × · · · × C r contain a spherical system ofgenerators, by straightforward iteration.4.7. For abelian groups G , conjugacy classes contain a single element, andthe algorithm can be improved.First, observe that Frobenius’ test is useless in this case: the product C × · · · × C r contains a single element ( g , . . . , g r ), so the condition Q g i = 1is best verified directly.Second, since refined passports contain a single element of G r , we effec-tively iterate through elements of G r . However, in a spherical system ofgenerators ( g , . . . , g r ) any element is determined by the others, so we caniterate through “short sequences” ( g , . . . , ˆ g j , . . . , g r ). Thus, we proceed asfollows.We fix m j in m = ( m , . . . , m r ) such that the number of elements of G with order m j is largest; then, we use a scheme analogous to Algorithm 4to iterate through ( r − g , . . . , ˆ g j , . . . , g r ) ∈ G r − with signature( m , . . . , ˆ m j , . . . , m r ), one for each Aut( G )-orbit. We then define g j as theinverse of g · · · ˆ g j · · · g r ; if g j has order m j , the ( g , . . . , g r ) is a candidatefor a spherical system of generators with signature m . At this point, we testcondition ( ∗ ) and, if it holds, whether the elements g , . . . , g r generate thegroup G . Algorithm 1:
Computing the signatures input : integers g ≥ , d ≥ output: the set of signatures S d,g Function S d,g ( d, g ) if d prime then return ∅ // (S3) cannot be satisfied S d,g ← ∅ for r satisfying (S1) do D ← { n ∈ N | ≤ n < d, n divides d } ; for m , . . . , m r ∈ D , m ≤ · · · ≤ m r do if ( m , . . . , m r ) satisfies (S4) and (S5) then insert ( m , . . . , m r ) in S d,g return S d,g Algorithm 2:
Find counter-examples of genus g with r ≥ input : an integer g ≥ output: counter-examples of genus g with r ≥ G )-orbit Function admissible( G , m ) O ← { ord( g ) | g ∈ G } ; if r = 4 and G abelian then return false; else if r > and G cyclic then return false; else if some m i is not in O then return false else if g > and some o ∈ O is greater than g − then return false // Lemma 3.1 else decompose the abelianization of G as Z /k Z ⊕ · · · ⊕ Z /k p Z ,with each k i dividing k i +1 ; if at least elements in ( k , . . . , k p ) are even then return false // G surjects over ( Z / Z ) (Corollary 3.13) if p ≥ r then return false // r − elements cannot generate G else if exists j such that k p ∤ lcm( m , . . . , ˆ m j , . . . , m r ) then return false // § return true // passed all tests for ≤ d ≤ g − do determine S d,g by Algorithm 1; for m = { m k , . . . , m r k k } in S d,g do for G group of order d do if admissible( G , m ) then for ≤ i ≤ k do A i ← { C ∈ C G | ord( C ) = m i } CounterExamplesIn ( G, { A r , . . . , A r k k } ) // Findcounter-examples for group G and signature m using Algorithm 4 OME EVIDENCE FOR THE COLEMAN-OORT CONJECTURE 19
Algorithm 3:
Determine whether a refined passport is a counter-example input : a group G , a refined passport ( C , . . . , C r ) output: true if the refined passport is a counter-example, falseotherwise Function
IsCounterExample( G , C , . . . , C r ) if ( C , . . . , C r ) passes Frobenius’ test and N = r − then for ( g , . . . , g r − ) in C × · · · × C r − do g r ← ( g · · · g r − ) − ; if g r ∈ C r and h g , . . . , g r − i = G then return true return false Algorithm 4:
Find counter-examples for fixed group and signature input :
A group G and a nonempty multiset { A n , . . . , A n k k } whereeach A i is a nonempty set of conjugacy classes of G andeach n i is a nonnegative integer output: Counter-examples obtained by choosing n i elements in each A i , one for each Aut( G )-orbit Function
CounterExamplesIn( G , { A n , . . . , A n k k } , S = ∅ , H = Aut( G ) ) // This is a recursive function using two argumentswith default values: S is a multiset of conjugacyclasses of G ; H is a subgroup of Aut( G ) acting oneach A i if k = 0 then if IsCounterExample( G , S ) then print G, S else if n k = 0 then CounterExamplesIn( G , { A n , . . . , A n k − k − } , S , H ) else if A k contains a single element a then CounterExamplesIn( G , { A n , . . . , A n k − k − } , S ∪ { a n k } , H ) else CounterExamplesWith( G , { A n , . . . , A n k − k − } , S , H , A k , n k ) Function
CounterExamplesWith( G , { A n , . . . , A n k k } , S , H , A , n ) // Helper function that iterates through subsets of A for ≤ h ≤ n do X ← { B ⊂ A | | B | = h } for one B in each H -orbit of X do K ← stabilizer of B for action of H on X CounterExamplesIn( G , { A n , . . . , A n k k , B n − h } , S ∪ B , K )This is clearly faster than a plain application of Algorithm 4, because an r -fold iteration is replaced by an ( r − that the same counter-example can appear more than once in the output, ifthis method is applied to cases where m j has multiplicity greater than one,say m j = m j +1 . Indeed, a counter-example ( g , . . . , g r ) can be obtainedby completing two different short sequences, namely ( g , . . . , b g j , . . . , g r ) and( g , . . . , d g j +1 , . . . , g r ). If the two short sequences lie in different Aut( G )-orbits, the output will contain two counter-examples in the Aut( G )-orbit of( g , . . . , g r ). References [1] E. Arbarello, M. Cornalba, and P. A. Griffiths.
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