L-functions associated with families of toric exponential sums
aa r X i v : . [ m a t h . N T ] J u l L -functions associated with families of toric exponential sums C. Douglas Haessig ∗ Steven SperberDecember 5, 2018
Abstract
We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponentialsums over a finite field. We construct a relative polytope with the aid of which we define a ring of coefficientsconsisting of p -adic analytic functions with polyhedral growth prescribed by the relative polytope. Using thiswe compute relative cohomology for such families and calculate sharp estimates for the relative Frobeniusmap. In applications one is interested in L -functions associated with linear algebra operations (symmetricpowers, tensor powers, exterior powers and combinations thereof) applied to relative Frobenius. Using methodspioneered by Ax, Katz and Bombieri we prove estimates for the degree and total degree of the associated L -function and p -divisibility of the reciprocal zeros and poles. Similar estimates are then established for affinefamilies and pure Archimedean weight families (in the simplicial case). Symmetric power L -functions and their variants have long been objects of study and have been valuable in manynumber theoretic applications. In the function field case, these L -functions are associated with families of varietiesor families of exponential sums defined over a base space S which is itself a variety over F q , the finite field of q = p a elements having characteristic p . For each closed point s of S , there is a zeta function or L -function for thefibre over this point. This is a rational function for each such s , and the collection A ( s ) of reciprocal zeros andpoles of this function is a finite set of algebraic integers. Interesting new L -functions may be created by takingEuler products as follows. Let A ( s ) ⊂ A ( s ) be an interesting well-chosen subset for each point s ∈ | S/ F q | , theset of closed points of S . For example, some choices include taking the subset of A ( s ) consisting of all p -adicunits, or all elements of A ( s ) having a fixed archimedean weight. Once chosen, we can form symmetric, tensor,or exterior powers (or combinations thereof) of the elements of A ( s ) and denote the resulting set by LA ( s ).Then the Euler product we are interested in has the general form L ( LA , S/ F q , T ) := Y s ∈| S/ F q | Y τ ( s ) ∈LA ( s ) (1 − τ ( s ) T deg ( s ) ) − , (1) ∗ Partially supported by NSF grant DMS-0901542 deg ( s ) := [ F q ( s ) : F q ] is the degree of the point.The p -adic study of symmetric power L -functions begins with Dwork [18], whose work was itself inspired byIhara [22] and Morita [27], who linked the Ramanujan-Petersson conjectures and the Weil conjectures. Implicitin their work were symmetric power L -functions for a suitable family of elliptic curves. In [18], Dwork explicitlyconsidered symmetric power L -functions associated with the Legendre family of elliptic curves, applying p -adiccohomology to obtain important information about this L -function. This study was continued by Adolphson[2] to obtain congruence information. Adolphson also considered symmetric power L -functions for the family ofelliptic curves with level three structure [3]. Based on Dwork’s p -adic cohomology theory of the Bessel function[16], Robba [28] p -adically studied the symmetric power L -functions of the family of Kloosterman sums. A similarstudy was made in Haessig [21] for cubic exponential sums.Symmetric, tensor, and exterior power L -functions have also been studied using ℓ -adic techniques going backat least to the work of Deligne. Katz [25] has studied symmetric power L -functions of families of elliptic curvesand their monodromy behavior. In recent work, Fu and Wan have obtained very detailed information on sym-metric power L -functions of hyper-Kloosterman sums [19] [20]. In [13], Haessig and Rojas-Le´on studied the k -thsymmetric power L -functions for a one-parameter family of exponential sums in one variable.In another direction, Wan [29] [30] [31] proved a conjecture of Dwork’s that unit root L -functions which comefrom geometry are p -adically meromorphic by relating the unit root L -function to symmetric, tensor, and exteriorpower L -functions via Adams operations and then employing a suitable p -adic limiting argument.In the present work, we consider a general family of nondegenerate toric exponential sums. Let¯ G ( x, t ) := ¯ f ( x ) + ¯ P ( x, t ) ∈ F q [ x ± , . . . , x ± n , t ± , . . . , t ± s ]where ¯ f ( x ) is nondegenerate with respect to ∆ ∞ ( ¯ f ), its Newton polyhedron at ∞ . We assume as well that themonomials in the x -variables in ¯ P ( x, t ) have strictly smaller (polyhedral) weight than the leading monomialsin ¯ f , and that the dim ∆ ∞ ( ¯ f ) = n . As a consequence, for each choice λ ∈ F ∗ sq , an F q -rational point of G sm where F q is an algebraic closure of F q , the Newton polyhedron at ∞ of ¯ G ( x, λ ) is nondegenerate with respectto ∆ ∞ ( ¯ G ( x, λ )) = ∆ ∞ ( ¯ f ). Let F q ( λ ) be the field of definition of λ and deg ( λ ) := [ F q ( λ ) : F q ] its degree. Let Cone (∆) be the cone over ∆ := ∆ ∞ ( ¯ f ), and M (∆) := Z n ∩ Cone (∆). Fix an additive character Θ of F q , and setΘ λ := Θ ◦ T r F q ( λ ) / F q . Let ¯ G λ ( x ) := ¯ G ( x, λ ). Define the exponential sums S r ( ¯ G λ , Θ , G nm / F q ( λ )) := X x ∈ ( F ∗ qrdeg ( λ ) ) n Θ λ ◦ T r F qrdeg ( λ ) / F q ( λ ) ¯ G λ ( x )and the associated L -function L ( ¯ G λ , Θ , G nm / F q ( λ ) , T ) := exp ∞ X r =1 S r ( ¯ G λ , Θ , G nm / F q ( λ )) T r r ! . S r ( λ ) and the associated L -functionsby L ( ¯ G λ , T ). L -functions of this type have been studied in [5] where it was shown that L ( ¯ G λ , T ) ( − n +1 is apolynomial of degree N := n ! vol ∆ ∞ ( ¯ f ) whose coefficients lie in the cyclotomic field Q ( ζ p ) of p -th roots of unity,and the reciprocal zeros are algebraic integers. Using the p -adic absolute value normalized at the fiber over λ byrequiring ord λ ( q deg ( λ ) ) = 1, then the lower bound for the Newton polygon of L ( ¯ G λ , T ) ( − n +1 , calculated using ord λ , is independent of λ and given in [5], as well as in (51) below. Denef and Loeser [12] have given a precisedescription of the distribution of archimedean weights for the reciprocal zeros of L ( ¯ G λ , T ) ( − n +1 . The results ofDenef-Loeser are independent of λ as well.We write then for each λ ∈ F ∗ sq , L ( ¯ G λ , T ) ( − n +1 = (1 − π ( λ ) T ) · · · (1 − π N ( λ ) T ) . Let A ( λ ) := { π i ( λ ) } Ni =1 . To fix ideas, we present some examples of L -functions of the form (1). The k -th tensorpower L -function of the toric family above is the Euler product L ( A ⊗ k , G sm / F q , T ) := Y λ ∈| G sm / F q | Y (1 − π i ( λ ) π i ( λ ) · · · π i k ( λ ) T deg ( λ ) ) − , where the inner product on the right runs over all k -tuples ( i , . . . , i k ) ∈ S k with S := { , , . . . , N } . Similarlythe k -th symmetric power L -function is given by L ( Sym k A , G sm / F q , T ) := Y λ ∈| G sm / F q | Y (1 − π ( λ ) i · · · π N ( λ ) i N T deg ( λ ) ) − where the inner product runs over N -tuples of non-negative integers ( i , . . . , i N ) satisfying i + · · · + i N = k .Another variant of interest focuses on the subset A ( λ ) ⊂ A ( λ ) consisting of the unique p -adic unit root, say π ( λ ), in A ( λ ). Then the k -th moment unit root L -function is defined by L unit ( k, ¯ G, G sm / F q , T ) := Y λ ∈| G sm / F q | (1 − π ( λ ) k T deg ( λ ) ) − . (2)We may also denote by W i ( λ ) the subset of A ( λ ) consisting of reciprocal zeros π ( λ ) having archimedean weightequal to i , that is, | π ( λ ) | C = q deg ( λ ) i/ . Then, we define L ( LW i , G sm / F q , T ) := Y λ ∈| G sm / F q | Y τ ( λ ) ∈LW i ( λ ) (1 − τ ( λ ) T deg ( λ ) ) − . We now state our main result for the full toric family A := ` λ ∈| G sm / F q | A ( λ ). Similar statements about otherfamilies may be found in the following sections. Section 4.1 looks at a family of affine exponential sums (in factit deals more generally with mixed toric and affine sums), Section 4.2 looks at a family of pure weight (in thearchimedean sense), and Section 4.3 looks at a p -adic unit root family. Let L N denote the cardinality of the set3 A ( λ ); this number is independent of the choice of λ . Let Γ ⊂ R s be the relative polytope of ¯ G , as defined asfollows (see also (19)). Let w be the polyhedral weight function defined by ∆ ∞ ( ¯ f ) in R n . DefineΓ := Convex hull in R s of the points { } ∪ (cid:26)(cid:18) − w ( µ ) (cid:19) γ ∈ Q s | ( γ, µ ) ∈ Supp ( ¯ P ) (cid:27) . (3)Let ˜ s denote the dimension of the smallest linear subspace of R s which contains Γ, and denote by vol (Γ) thevolume of Γ in this linear subspace with respect to Haar measure normalized so that a fundamental domain of theinteger lattice in the subspace has unit volume. Lastly, we define the order |L| := r of a linear algebra operation L as the least positive integer r such that L is a quotient of an r -fold tensor product. Theorem 1.1.
For each linear algebra operation L , the L -function L ( LA , G sm / F q , T ) is a rational function: L ( LA , G sm / F q , T ) ( − s +1 = Q Ri =1 (1 − α i T ) Q Sj =1 (1 − β j T ) ∈ Q ( ζ p )( T ) . Furthermore, writing this in reduced form ( α i = β j for every i and j ):(a) The reciprocal zeros and poles α i and β j are algebraic integers. For each reciprocal pole β j there is areciprocal zero α k j and a positive integer m j such that β j = q m j α k j .(b) The degree R − S of the L -function as a rational function is bounded as follows. If ˜ s < s then R = S , elseif ˜ s = s then ≤ R − S ≤ s ! vol (Γ) L N. (c) The total degree R + S of the L -functions is bounded above by R + S ≤ L N · ˜ s ! vol (Γ) · ˜ s +(1+ s ) n |L| (1 + 2 s ) s . Next, we consider the L -function defined over affine s -space. To this end, we assume ¯ P ( x, t ) ∈ F q [ x ± , . . . , n ± n , t , . . . , t s ].Set M (Γ) := Z s ∩ Cone (Γ), where
Cone (Γ) is the union of all rays from the origin through Γ. With w Γ the poly-hedral weight function defined by Γ in R s , define w (Γ) := min { w Γ ( u ) | u ∈ M (Γ) ∩ Z s ≥ } . Let A ⊂ { , , . . . , s } . Let ¯ G A be the polynomial obtained from ¯ G by setting t i = 0 for each i ∈ A . In preciselythe same manner as Γ, let Γ A be the relative polytope of ¯ G A and define its volume vol (Γ A ) with respect to Haarmeasure normalized so that a fundamental domain of the integer lattice in the smallest subspace containing Γ A has unit volume. Theorem 1.2.
Suppose ¯ G ∈ F q [ x ± , . . . , x ± n , t , . . . , t s ] . For each linear algebra operation L , the L -function ( LA , A s / F q , T ) is a rational function: L ( LA , A s / F q , T ) ( − s +1 = Q Ri =1 (1 − α i T ) Q Sj =1 (1 − β j T ) ∈ Q ( ζ p )( T ) . Writing this in reduced form ( α i = β j for every i and j ):(a) The reciprocal zeros and poles satisfy ord q ( α i ) , ord q ( β j ) ≥ w (Γ) . (b) The degree is bounded by − X A ⊂{ , ,...,s }| A | odd ( s − | A | )! vol (Γ A ) ≤ R − S ≤ X A ⊂{ , ,...,s }| A | even ( s − | A | )! vol (Γ A ) . (c) The total degree is bounded by R + S ≤ ˜ s +(1+ s ) n |L| s L N · ˜ s ! vol (Γ)Lastly, we look at the case of an affine family over an affine base. Suppose ¯ G now is a polynomial in F q [ x , . . . , x n , t , . . . , t s ], and ¯ f is convenient and nondegenerate. For each λ ∈ | A s / F q | , let ˜ A ( λ ) = { π i ( λ ) } ˜ Ni =1 bethe set of reciprocal zeros of L ( ¯ G λ , Θ , A n / F q ( λ ) , T ) ( − n +1 . Define w (∆) for ∆ = ∆ ∞ ( ¯ f ) in a similar way to thatof w (Γ), so that w (∆) = min { w ( γ ) | γ ∈ M (∆) ∩ Z n ≥ } . Theorem 1.3.
Suppose the conditions of the previous paragraph. For each linear algebra operation L , the L -function L ( L ˜ A , A s / F q , T ) is a rational function: L ( L ˜ A , A s / F q , T ) ( − s +1 = Q (1 − α i T ) Q (1 − β j T ) ∈ Q ( ζ p )( T ) . Writing this in reduced form, then ord q ( α i ) and ord q ( β j ) ≥ w (Γ) + w (∆) L ˜ N . We note that the upper bound on the degree in Theorem 1.1 (b) and the lower bound on the p -adic order ofthe roots in Theorem 1.2 (a) are sharp in the sense that there are examples where the bounds are obtained (e.g.[21] and [13]).When we work with proper subsets A ⊂ A , we in general do not expect rationality of the L -function, forexample in the case when A is the unit root family. As mentioned earlier, Wan’s proof of Dwork’s conjectureuses a p -adic limiting argument of L -functions associated to Adams operations. Since Adams operations may beviewed as a virtual linear algebra operation, we may apply the lower bound in Theorem 1.2(a) to this sequence of L -functions to obtain a similar result for the unit root L -function (2). This is discussed in Sections 2.2 and 4.3.5e view the main contributions of the present study to be the discovery of the role played by the relativepolytope Γ for very general families of nondegenerate toric exponential sum. Previous p -adic studies have mainlybeen one parameter families in which the parameter appears linearly. In the present study we remove theserestrictions. We are able nevertheless to compute relative cohomology, and the relative polytope provides asufficiently good weight function so that the general results obtained are sharp in cases where the L -functionshave previously been computed. In other earlier work (see [1] and [5]), understanding the weight function was anessential step in enabling the calculation of p -adic cohomology. It is our hope that the relative polytope providesa similar key step here. In a future article, we intend to treat families of Kloosterman-like sums, including thecalculation of the relevant p -adic cohomology. It should be noted that while our main application has beento families of toric nondegenerate exponential sums these results have broader application to σ -modules withpolyhedral growth, the content of which is in Section 2. Acknowledgment:
We thank Nick Katz for providing the ℓ -adic proof of an upper bound for the p -adicorder of eigenvalues of Frobenius used in Section 3. L -functions Let Γ denote a fixed polytope in R s with rational vertices which contains the origin, perhaps on its boundary.Let ˜ s denote the dimension of the smallest linear space in R s containing Γ. We will assume ˜ s ≥
1. Let
Cone (Γ)be the union of all rays from the origin through Γ and set M (Γ) := Z s ∩ Cone (Γ). For each u ∈ M , we define theweight w ( u ) of u as the smallest nonnegative real number such that u is in the dilation w ( u )Γ. Since the valuesof w on M (Γ) may be described using rational linear forms coming from a finite number of top dimensional facesof the polytope, there exists a positive integer D = D (Γ) such that w ( M (Γ)) ⊂ (1 /D ) Z ≥ . With q = p a , let Q q denote the unramified extension of Q p of degree a , and Z q its ring of integers.With an eye toward obtaining p -adic estimates for the Frobenius below, we fix π , a zero of the series P ∞ j =0 y p j /p j having ord p π = 1 / ( p − Q q ( ζ p ) of Q p ( ζ p ). Thesefields have ring of integers respectively Z q [ π ] and Z p [ π ]. At various points in the exposition we will want to taketotally ramified extensions of Q q ( ζ p ) and Q p ( ζ p ). We will accomplish this by adjoining an appropriate root ˜ π of π , say ˜ π = π / ˜ D for some positive integer ˜ D . The Frobenius automorphism σ of Gal ( Q q / Q p ) is extended to Gal ( Q q (˜ π ) / Q p (˜ π )) by setting σ (˜ π ) = ˜ π . Let us take then K = Q q (˜ π ) and assume the ramification index of K/ Q p is e .Denote by Z q [˜ π ] the ring of integers of K . Using the multi-index notation t u := t u · · · t u s s , where u =( u , . . . , u s ), we define the overconvergent power series ring O † Γ as O † Γ := X u ∈ M (Γ) a u t u | a u ∈ Z q [˜ π ] , lim inf w ( u ) →∞ ord π ( a u ) w ( u ) > . Let A := ( A i,j ) i,j ∈ I be a square (possibly infinite) matrix with entries in O † Γ . We assume the index set is6ountable (or finite) and we take it to be I = { , , , . . . } . We will assume that A is nuclear , meaning hereits columns tend to zero p -adically (i.e. sup i | A i,j | → j → ∞ ). When A is a finite square matrix it isautomatically nuclear since the nuclear condition is vacuous. Let ˆ t ∈ ( Q ∗ p ) s be the Teichm¨uller lifting of a point¯ t ∈ ( F ∗ q ) s , and let deg (¯ t ) := [ F q (¯ t ) : F q ]. We extend σ to an automorphism of O † Γ by acting on the coefficients ofthe power series. Define the matrix B ( t ) := A σ a − ( t p a − ) · · · A σ ( t p ) A ( t ) . Following Dwork [15], we may associate to B the L -function L ( B, G sm / F q , T ) := Y ¯ t ∈| G sm / F q | det (1 − B (ˆ t q deg (¯ t ) − ) · · · B (ˆ t q ) B (ˆ t ) T deg (¯ t ) ) ∈ T Z q [ π ][[ T ]] , (4)where the product runs over all closed points of the algebraic torus G sm over F q . This may be generalized sothat the product runs over closed points of an algebraic variety; of particular interest is when the polytope Γ lieswithin the first quadrant R s ≥ so that we may define the L -function over affine s -space A s .As it stands, L ( B, G sm / F q , T ) is not in general a rational function. In fact, it is not even p -adic meromorphic ingeneral (see [32, Theorem 1.2]). However, we will either insist on its rationality or assume a uniform overconvergentcondition (5) which will guarantee p -adic meromorphy of the L -function.It is useful to measure the nuclear condition on A as follows. For each integer j ≥
0, let d j be the smallestnonnegative integer such that A i,j ′ ≡ π j ) for all j ′ ≥ d j , where we say A ij ≡ π j ) if ˜ π j dividesevery coefficient of the series A ij = A ij ( t ). That is, A i,d j , A i,d j +1 , A i,d j +2 , . . . is divisible by ˜ π j . Note that d = 0.Define h j := d j +1 − d j . This means that the first h columns of A are divisible by at least ˜ π = 1, the next h columns are divisible by at least ˜ π , the next h columns by ˜ π , and so forth. Next, we need a function which willprovide us with how divisible the j -th column is according to these h i . Set s (0) := 0, and for each integer j ≥ s ( j ) denote the largest integer such that j ≥ d s ( j ) . That is, s ( j ) = ℓ , where ℓ is the minimum integer suchthat j ∈ [0 , h + h + · · · + h ℓ ]. This means that column j of the matrix A is divisible by at least ˜ π s ( j ) .Now let A ( t ) be a square matrix indexed by I (possibly infinite) with entries series in O † Γ with coefficients in K . Let b ∈ R > , ρ ∈ R . Dwork defined spaces L ( b ; ρ ) := X u ∈ M (Γ) c u t u | c u ∈ Q q (˜ π ) , ord p ( c u ) ≥ bw ( u ) + ρ L ( b ) := [ ρ ∈ R L ( b ; ρ ) . Write A ( t ) = P u ∈ M (Γ) a u t u where a u = ( a u ( i, j )) i,j ∈ I is a matrix with coefficients in Z q [˜ π ]. When I is infinitewe shall assume that lim inf i,j ∈ I and w ( u ) →∞ ord p ( a u ( i, j )˜ π − s ( j ) ) w ( u ) > , (5)7 uniform overconvergent condition. We note that w ( u ) provides the polyhedral growth of the coefficients a u ( i, j )with respect to t u while ˜ π s ( j ) describes the weight placed on each column index j . When A satisfies (5), includingthe case when I is finite, then there exists a positive real number b in which all entries of A lie within the Dworkspace L ( b ). Furthermore, there exists a real number ρ such that A i,j ∈ L ( b ; s ( j ) e + ρ ) for all i, j ∈ I . In this case,we may define the matrix A ′ := p − ρ A whose entries A ′ i,j ∈ L ( b ; s ( j ) /e ) for all i, j ∈ I , and L ( A ′ , G sm / F q , T ) = L ( p − ρ A, G sm / F q , T ) = L ( A, G sm / F q , p − ρ T ) . Thus, we may at various points throughout the paper assume that A has been normalized such that ρ = 0, orequivalently, h = 0.Write B ( t ) = P u ∈ M (Γ) b u t u . Define F B as the matrix ( b qu − v ) ( u,v ) , where u and v run over M (Γ), and we set b qu − v = 0 if qu − v M (Γ). That is, the ( u, v ) block entry of F B is the matrix b qu − v . Note, even if B is a finitedimensional matrix, F B is infinite dimensional. Dwork’s trace formula [32, Lemma 4.1] says L ( B, G sm / F q , T ) ( − s +1 = s Y i =0 det (1 − q i F B T ) ( − i ( si )= det (1 − F B T ) δ s , (6)where δ sends an arbitrary function g ( T ) to the quotient g ( T ) /g ( qT ). Since the entries a u satisfy (5), so dothe entries b u ; in particular, if A i,j ∈ L ( b ) then B i,j ∈ L ( b/p a − ). Consequently, the Fredholm determinant det (1 − q i F B T ) is p -adic entire by [32, Proposition 3.6], and thus L ( B, G sm / F q , T ) is p -adic meromorphic on C p by (6). 8 .1 Main theorems on general L -functions In this section we prove two main results about L -functions of overconvergent matrices defined over O † Γ . We recallthe following definitions and associated data from the previous section:Γ , a rational polytope in R s with volume vol (Γ)˜ s, the dimension of the smallest linear space in R s containing Γ Cone (Γ) , the cone in R s over Γ M (Γ) := Z s ∩ Cone (Γ) K = Q q (˜ π ) , with ord p (˜ π ) = 1 /eA = ( A i,j ) i,j ∈ I , a matrix with entries satisfying A i,j ∈ L ( b ; s ( j ) /e ) for some fixed b ∈ Q > B, a matrix defined by B ( t ) := A σ a − ( t p a − ) · · · A σ ( t p ) A ( t ) w (Γ) := min { w ( u ) | u ∈ M (Γ) ∩ Z s> } ord p ( A ) := min { s ( i ) /e | i ∈ I } . It is convenient to assume that the field K is sufficiently ramified so that the denominator of b and D (Γ)divide e . This does not for example change ord p ( A ). Suppose A , and hence B , are N × N matrices and write dim ( B ) := N . Suppose L ( B, G sm / F q , T ) is a rational function: L ( B, G sm / F q , T ) ( − s +1 = Q Ri =1 (1 − α i T ) Q Sj =1 (1 − β j T ) (7)written in reduced form (i.e. α i = β j for every i and j ). Let Z := { α , . . . , α R } be the multiset of reciprocal zerosand P := { β , . . . , β S } the multiset of reciprocal poles of (7). Two elements γ , γ ∈ Z ∪ P are said to be q -related if there is an integer τ such that γ = q τ γ . This defines an equivalence relation on Z ∪ P . Let E := { E , . . . , E ℓ } be the set of equivalence classes. Focussing on a fixed E ∈ E , define L E ( B, G sm / F q , T ) ( − s +1 := Q α ∈ Z ∩ E (1 − αT ) Q β ∈ P ∩ E (1 − βT ) , and if R E denotes the cardinality of Z ∩ E and S E the cardinality of P ∩ E , then we may order the reciprocalzeros and poles giving L E ( B, G sm / F q , T ) ( − s +1 = Q R E i =1 (1 − α i T ) Q S E j =1 (1 − β j T ) . Clearly, L ( B, G sm / F q , T ) = Y E ∈E L E ( B, G sm / F q , T ) . Lemma 2.1.
Under the conditions assumed above, then1. for each E ∈ E , R E − S E ≥ . (Consequently, the degree R − S of L ( B, G sm / F q , T ) ( − s +1 as a rational unction is nonnegative.)2. For each E ∈ E , there is a choice γ E ∈ { α , . . . , α R E } such that there are nonnegative integers { m i } R E i =1 andstrictly positive integers { n j } S E j =1 with α i = q m i γ E and β j = q n j γ E . Proof.
The following proof illustrates Bombieri’s method [9, Section 4, p.83]. For γ ∈ Z ∪ P we write H ( γ ) := ∞ Y m =0 (1 − q m γT ) c ( m ) where c ( m ) := (cid:0) m + s − s − (cid:1) . The δ -structure (6) implies det (1 − F B T ) = Q α ∈ Z H ( α ) Q β ∈ P H ( β ) , and this is p -adically entire as noted above. Note that for two elements γ , γ ∈ Z ∪ P , γ and γ are q -related ifand only if H ( γ ) and H ( γ ) have a factor in common. As a consequence, if we write D E ( T ) := Q α ∈ Z ∩ E H ( α ) Q β ∈ P ∩ E H ( β ) , then det (1 − F B T ) = Y E ∈E D E ( T )and each D E ( T ) is p -adically entire. If we fix β ∈ E ∩ P so that β = q m j β j with m j ≥ j = 2 , . . . , S E , thenwe also have β = q t i α i with t i a non-zero integer for i = 1 , . . . , S E . The precise divisibility of (1 − q m β T ) in Q R E i =1 H ( α i ), for m sufficiently large, is P R E i =1 c ( m + t i ) which grows with m like a polynomial of the form R E m s − ( s − +(lower order terms). Similarly, the precise divisibility of (1 − q m β T ) in Q S E j =1 H ( β j ) is c ( m ) + P S E j =2 c ( m + m j )which grows like a polynomial of the form S E m s − ( s − + (lower order terms). Since D E ( T ) is p -adically entire, anyfactor (1 − q m β T ) must have nonnegative exponent in Q R E i =1 H ( α i ) / Q S E j =1 H ( β j ) so that for m sufficiently large,we must have R E − S E ≥
0. This completes the proof of the first part of the theorem.The proof of the second part is simpler. Note that D E ( T ) is entire so that each factor (1 − βT ) must divide Q R E i =1 H ( α i ), as a consequence β j = q m j α k ( j ) for some m j ≥ ≤ k ( j ) ≤ R E . The result then follows bychoosing γ E among { α , . . . , α R E } so that α i = q m i γ E for nonnegative integers { m i } R E i =1 .We note that it is an interesting question to determine an upper bound for the degree of L E ( B, G sm / F q , T ),as well as an estimate for the number of equivalence classes in E . Theorem 2.2.
Suppose A , and hence B , are N × N matrices and write dim ( B ) := N . In parts (a) through (c) elow, we assume that L ( B, G sm / F q , T ) is a rational function: L ( B, G sm / F q , T ) ( − s +1 = Q Ri =1 (1 − α i T ) Q Sj =1 (1 − β j T ) written in reduced form (i.e. α i = β j for every i and j ). In part (d), we assume condition (5) so that this L -function is p -adic meromorphic. Then,a) The reciprocal zeros and poles α i and β j are algebraic integers. Also, for each reciprocal pole β j there is areciprocal zero α k j and a positive integer m j such that β j = q m j α k j .b) If ˜ s < s then R = S , else if ˜ s = s then ≤ R − S ≤ (cid:18) b ( p − (cid:19) s · s ! vol (Γ) · dim ( B ) c) Let k be the smallest positive integer such that ord q α i and ord q β j ≤ k for all i and j . Then, with ρ :=min { s, k } , we have R + S ≤ dim ( B ) · ˜ s ! vol (Γ) · ˜ s + b ( p − (1+ s )( k − ρ ) (1 + 2 b ( p − (1+ s ) ) ρ . d) Suppose Γ ⊂ R s ≥ and b ( p − ≤ . Write L ( B, A s / F q , T ) = Q (1 − α i T ) Q (1 − β j T ) . Here, A , and hence B , may be infinite dimensional, in which case α i and β j → p -adically. Then ord q α i and ord q β j ≥ b ( p − w (Γ) + ord p ( A ) for all i and j . A lower bound is also given when b ( p − > .Proof of Theorem 2.2, part (a). That the reciprocal roots are algebraic integers follows directly from Dwork’sargument found in Bombieri’s paper [9, Section 4, p.82]. The second part follows from Lemma 2.1.The proof of parts (b) , (c) , and (d) will require various lemmas. We begin by obtaining a lower bound on the q -adic Newton polygon of det (1 − F B T ). With this purpose in mind, we define a p -adic Banach space C ( b, I ) overthe field K := Q p (˜ π ), having orthonormal basis { γ w ( u ) b t u e i } u ∈ M (Γ) ,i ∈ I where ord p ( γ b ) = b . Thus C ( b, I ) := ξ = X i ∈ I,u ∈ M (Γ) c ( u, i ) γ w ( u ) b t u e i | c ( u, i ) ∈ K , c ( u, i ) → u, i ) → ∞ . As usual, the norm on C ( b, I ) is given by | ξ | := sup {| c ( u, i ) | : ( u, i ) ∈ M (Γ) × I } . The map ψ p ◦ A ( t ) acts on ξ ∈ C ( b, I ) via (recall the notation a v ( i, j ) from the sentence before equation (5))( ψ p ◦ A ( t )) ξ = X i ∈ I,ℓ ∈ M (Γ) X j ∈ I,u + v = pℓ a v ( i, j ) c ( u, j ) γ w ( u ) − w ( ℓ ) b γ w ( ℓ ) b t ℓ e i . A := σ − ◦ ψ p ◦ A ( t ) is a completely continuous endomorphism of C ( b, I ) over K , semi-linear with respectto σ − over K . The map Φ B := ψ q ◦ B ( t ) is a completely continuous endomorphism of C ( b, I ) over K satisfyingΦ B = Φ aA . Let B := { t u e i | ( u, i ) ∈ M (Γ) × I } . Then the matrix of Φ B computed with respect to the basis B is F B . As is well-known, a completely continuous endomorphism has a well-defined Fredholm determinant det K (1 − Φ B T ). This may be computed using the matrix of Φ B with respect to B .Let { η , . . . , η a } ⊂ Z q be a lifting of a basis of F q over F p . Then { η j } aj =1 is a basis of Q q over Q p such that forevery ξ ∈ Q q , writing ξ = P aj =1 ξ j η j with ξ j ∈ Q p , we have ord p ( ξ ) = min { ord p ( ξ j ) } . The Fredholm determinantof Φ A as a completely continuous endomorphism of the K -space C ( b, I ) may be calculated from the matrix of Φ A with respect to the basis B ′ := { η j t u e i : 1 ≤ j ≤ a, u ∈ M (Γ) , i ∈ I } . The relation the between det K (1 − Φ A T )and det K (1 − Φ B T ) is given by the following lemma. Lemma 2.3. (cf. [14, Lemma 7.1]) Taking the graph of the p -adic Newton polygon of det K (1 − Φ A T ) andrescaling the abscissa and ordinate by /a yields the q -adic Newton polygon of det K (1 − Φ B T ) .Proof. For convenience, write G ( T ) := det K (1 − Φ B T ). Then det K (1 − Φ B T ) = N orm
K/K G ( T )= G σ a − ( T ) · · · G σ ( T ) G ( T ) . Next, det K (1 − Φ B T a ) = det K (1 − Φ aA T a )= Y ζ a =1 det K (1 − ζ Φ A T ) . Thus, G σ a − ( T a ) · · · G σ ( T a ) G ( T a ) = Y ζ a =1 det K (1 − ζ Φ A T ) . (8)Let N denote the number of reciprocal roots of G ( T ) with slope ord q = m , which means ord p = am . Then thelefthand-side of (8) has ( aN ) a number of reciprocal roots of ord p = m . Consequently, since ζ does not affect theNewton polygon of det K (1 − ζ Φ A T ), we see that det K (1 − Φ A T ) has aN reciprocal zeros of ord p = m . Thelemma follows.We proceed now to an estimate for the p -adic Newton polygon of det K (1 − Φ A T ). Recall, A ( t ) = ( A i,j )satisfies A i,j ∈ L ( b ; s ( j ) /e ) for every j , with b a positive rational number. Let d be the smallest positive integersuch that b ( p − w ( u ) + s ( i ) /e ∈ d Z for all u ∈ M (Γ) and i ∈ I . Note, this means db ( p −
1) and ds ( i ) /e arenonnegative integers for all i ∈ I . Define W ( j ) := (cid:26) ( u, i ) ∈ M (Γ) × I | b ( p − w ( u ) + s ( i ) e = jd (cid:27) . emma 2.4. The p -adic Newton polygon of det K (1 − Φ A T ) lies on or above the lower convex hull in R of thepoints (0 , and a n X j =0 W ( j ) , ad n X j =0 jW ( j ) n = 0 , , , . . . . Consequently, from Lemma 2.3, the q -adic Newton polygon of det (1 − F B T ) lies on or above the lower convex hullin R of the points (0 , and n X j =0 W ( j ) , d n X j =0 jW ( j ) n = 0 , , , . . . . (9) Proof.
Write A ( t ) = P u ∈ M (Γ) a u t u , where each a u = ( a u ( i, j )) i,j ∈ I is a matrix. For each basis element e i , write a u e i = P j ∈ I a u ( i, j ) e j with a u ( i, j ) ∈ K . We now compute the matrix of Φ A with respect to the basis B ′ . For η l t u e i ∈ B ′ , Φ A ( η l t u e i ) = σ − ◦ ψ p ( A ( t ) η l t u e i )= σ − ( η l ) · ψ p X v ∈ M (Γ) X j ∈ I a v ( i, j ) t v + u e j = σ − ( η l ) · ψ p X v ∈ M (Γ) X j ∈ I a v ( i, j ) e j t u + v = σ − ( η l ) · ψ p X r ∈ M (Γ) X j ∈ I a r − u ( i, j ) e j t r = σ − ( η l ) · X r ∈ M (Γ) X j ∈ I a pr − u ( i, j ) e j t r . For each i and j , write a u ( i, j ) = P ak =1 a u ( i, j ; k ) η k , with a u ( i, j ; k ) ∈ K . Continuing the above calculation,Φ A ( η l t u e i ) = σ − ( η l ) · X r ∈ M (Γ) X j ∈ I a X k =1 a pr − u ( i, j ; k ) η k t r e j . Next, for each l and k , write σ − ( η l ) η k = P am =1 b ( l, k ; m ) η m with b ( l, m ; k ) ∈ Z p . Then, continuing the calculationΦ A ( η l t u e i ) = X r ∈ M (Γ) X j ∈ I a X k,m =1 a pr − u ( i, j ; k ) b ( l, k ; m ) η m t r e j . Hence, Matrix of Φ A with respect to the basis B ′ is a X k =1 a pr − u ( i, j ; k ) b ( l, k ; m ) ! ( l,u,i ) , ( m,r,j ) . Set d ( l, u, i ; m, r, j ) := P ak =1 a pr − u ( i, j ; k ) b ( l, k ; m ). Then det K (1 − Φ A T ) = ∞ X m =0 c m T m c m := ( − m X X τ ∈ S m sgn ( τ ) m Y z =1 d ( l ( z ) , u ( z ) , i ( z ) ; l ( τ ( z )) , u ( τ ( z )) , i ( τ ( z )) )where the first sum runs over all m number of triples ( l (1) , u (1) , i (1) ) , . . . , ( l ( m ) , u ( m ) , i ( m ) ) of distinct elementsof the set { , , . . . , a } × M (Γ) × I , and S m is the symmetric group on the letters { , , . . . , m } . Recall that ord p ( a u ( i, j )) ≥ bw Γ ( u ) + s ( i ) e , and so by construction of the basis { η j } , the same holds true for each a u ( i, j ; k ).Since ord p ( b ( l, k ; m )) ≥
0, we have ord p ( c m ) ≥ min distinct ( l,u,i ) min τ ∈ S m m X z =1 (cid:18) bw Γ ( pu ( τ ( z )) − u ( z ) ) + s ( i τ ( z ) ) e (cid:19) ≥ min distinct ( l,u,i ) ( m X z =1 (cid:18) b ( p − w Γ ( u ( z ) ) + s ( i ( z ) ) e (cid:19)) . It follows that the p -adic Newton polygon of det K (1 − Φ A T ) lies on or above the lower convex hull in R of thepoints (0 ,
0) and a n X j =0 W ( j ) , ad n X j =0 jW ( j ) n = 0 , , , . . . . Proof of Theorem 2.2, part (b).
Bombieri’s argument [9, Section IV] demonstrates that using the lower bound(9) and the Dwork trace formula (6), one may obtain the inequality deg L ( B, G sm , T ) ( − s +1 ( s + 1)! x s +1 + O ( x s ) ≤ X jd ≤ x (cid:18) x − jd (cid:19) W ( j ) . As we show below, the righthand side may be asymptotically approximated by X jd ≤ x (cid:18) x − jd (cid:19) W ( j ) = dim ( B ) vol (Γ)(˜ s + 1)( b ( p − ˜ s x ˜ s +1 + O ( x ˜ s ) , (10)which proves the result since ˜ s ≤ s . The proof of this estimate is a modification of an argument of [1, § W ( j ) = X i ∈ I W i ( j ) (11)where W i ( j ) := { u ∈ M (Γ) | b ( p − w ( u ) + s ( i ) /e = j/d } . For convenience, set b ′ := b ( p −
1) and c i := s ( i ) /e . Recall that w ( M (Γ)) ⊂ (1 /D ) Z ≥ . Now, if u satisfies b ′ w ( u ) + c i = j/d then there exists a nonnegative integer j ′ such that w ( u ) = j ′ /D such that b ′ j ′ /D + c i = j/d .14ence, for each fixed i , X jd ≤ x W i ( j ) = X j ′ D ≤ x − cib ′ { u ∈ M (Γ) | w ( u ) = j ′ /D } = vol (Γ) (cid:16) xb ′ (cid:17) ˜ s + O ( x ˜ s − ) , where we have used the argument in [1, §
4] for the second equality. A similar calculation gives X jd ≤ x (cid:18) jd (cid:19) W i ( j ) = X j ′ D ≤ x − cib ′ (cid:18) b ′ j ′ D + c i (cid:19) { u ∈ M (Γ) | w ( u ) = j ′ /D } = ˜ s vol (Γ)˜ s + 1 (cid:16) xb ′ (cid:17) ˜ s +1 + O ( x ˜ s ) . Then, (10) follows by combining these estimates with (11).We now move on to an upper bound for the total degree. Like the degree, the proof is a modification of anargument of Bombieri’s [10]. However, we will follow the argument in [1] since it applies to growth conditionsdefined by polytopes.
Proof of Theorem 2.2, part (c).
In fact, we prove the slightly stronger inequality R + S ≤ dim ( B ) · ˜ s ! vol (Γ) · ˜ s − sdb ( p − + b ( p − (1+ s )( k − ρ ) (1 + 2 b ( p − (1+ s ) ) ρ . (12)Inequality (12) follows from an almost identical argument to that in [1, §
5] once one has the following rationalityresult of the Poincar´e series P ∞ N =0 W ( N ) T N : ∞ X N =0 W ( N ) T N = Q ( T )(1 − T db ( p − ) ˜ s , (13)where Q ( T ) is a polynomial with nonnegative integer coefficients and special value Q (1) = dim ( B ) · ˜ s ! vol (Γ). Weprove this as follows. Define W ′ ( N ) := { u ∈ M (Γ) | w ( u ) = N/D } . The Poincar´e series of this, by [1, Lemma 5.1], satisfies ∞ X N ′ =0 W ′ ( N ′ ) T N ′ = P ( T )(1 − T D ) ˜ s with P ( T ) a polynomial in nonnegative integral coefficients, degree at most ˜ sD , and special value P (1) = ˜ s ! vol (Γ).Using (11), observe that ∞ X N =0 W ( N ) T N = X i ∈ I ∞ X N =0 W i ( N ) T N . For convenience, set b ′ := b ( p −
1) and c i := s ( i ) /e . Notice that W i ( N ) is a positive integer if and only if there15s a u ∈ M (Γ) that satisfies b ′ w ( u ) + c i = N/d . By defining the integer N ′ by w ( u ) = N ′ /D , we see that W i ( N ) = W ′ ( N ′ ) when Nd = b ′ · N ′ D + c i . Thus, X i ∈ I ∞ X N =0 W i ( N ) T N = X i ∈ I ∞ X N ′ =0 W ′ ( N ′ ) T d (( b ′ N ′ /d )+ c i ) = P ( T ) P i ∈ I T dc i (1 − T db ′ ) ˜ s . Note, by the discussion before Lemma 2.4, db ′ and dc i are nonnegative integers. Denoting the numerator of thisquotient by Q ( T ) proves (13).We remark that a more modern and accessible presentation of Ehrhart’s theory on polytopes (e.g. rationalityof the Poincar´e series mentioned in the above proof) may be found in [8, Chapter 3].Assume now that Γ lies entirely in R s ≥ so that the L -function may be defined over affine s -space A s . Define S ( B ) := X ¯ t ∈ F sq T r ( B ( t )) , where t is an s -tuple in Z sq whose coordinates are the Teichm¨uller representative of ¯ t (or zero). Theorem 2.5.
Suppose Γ lies entirely in R s ≥ . Here A may be infinite dimensional. Assume b ( p − ≤ . Then ord q S ( B ) ≥ b ( p − w (Γ) + ord p ( A ) . (14) A lower bound is also given if b ( p − > .Proof. The underlying idea of the following goes back to Katz [23]. Write B ( t ) = P u ∈ M (Γ) b u t u . For C ⊂{ , , . . . , s } , define the matrix B C by B C := X u ∈ M (Γ) u i > i ∈ C b u t u . We use the following version of the Dwork trace formula [32, Lemma 4.3]: L ( B, A s / F q , T ) = Y C ⊂{ , ,...,s } det (1 − q s −| C | F B C T ) ( − s −| C | . Hence, S ( B ) = X C ⊂{ , ,...,s } ( − | C | q s −| C | T r ( F B C ) , and since q = p a , we have ord q S ( B ) ≥ min C ⊂{ , ,...,s } { s − | C | + ord q ( T r ( F B C )) } .
16e now estimate ord q ( T r ( F B C )). Let E C denote the set of u ∈ M (Γ) with u i > i ∈ C . Define W C ( j ) := (cid:26) ( u, i ) ∈ E C × I | b ( p − w ( u ) + s ( i ) e = jd (cid:27) . (15)Using a similar argument to Lemma 2.4, the q -adic Newton polygon of det K (1 − F B C T ) lies on or above thepoints (0 ,
0) and k X j =0 W C ( j ) , d k X j =0 jW C ( j ) k = 0 , , , . . . Let k be the first nonnegative integer such that W C ( k ) = 0. Then the first point away from the origin in theabove sequence is ( W C ( k ) , k d W C ( k )), and thus, since det K (1 − F B C T ) = 1 − T r ( F B C ) T + O ( T ), we obtainthe inequality ord q ( T r ( F B C )) ≥ k d = min ( u,i ) ∈ E C × I (cid:26) b ( p − w ( u ) + s ( i ) e (cid:27) . Now, using [4, Lemma 4.5] for the second inequality below, we see that ord q S ( B ) ≥ min C ⊂{ , ,...,s } (cid:26) s − | C | + min ( u,i ) ∈ E C × I (cid:26) b ( p − w ( u ) + s ( i ) e (cid:27)(cid:27) = min C ⊂{ , ,...,s } (cid:26) s − | C | + b ( p −
1) min u ∈ E C { w ( u ) } (cid:27) + ord p ( A ) ≥ min C ⊂{ , ,...,s } { s − | C | + b ( p −
1) ( w (Γ) − ( s − | C | )) } + ord p ( A )= b ( p − w (Γ) + ord p ( A ) if b ( p − ≤ s (1 − b ( p − b ( p − w (Γ) + ord p ( A ) if 1 < b ( p − , taking | C | = s in the former and | C | = 0 in the latter. Proof of Theorem 2.2, part (d).
For convenience, set κ := b ( p − w (Γ) + ord p ( A ). With S k ( B ) := X ¯ t ∈ ( F qk ) s T r ( B ( t )) , it follows from Theorem 2.5 that ord q ( S k ( B )) ≥ kκ for every k . Using the same idea as in [7], this is equivalentto every reciprocal zero and pole of the L -function of B having p -adic order at least κ . σ -modules Using freely the terminology of [29] and [30], we will demonstrate that part (d) of Theorem 2.2 may be applied toobtain a similar result for unit root σ -modules. In Section 4.3, we will apply the following result to a particular unitroot L -function coming from the nondegenerate toric family in Section 3. Let ( M, φ ) be a finite rank, ordinary,nuclear σ -module. (Here, ordinary means that the fiber-by-fiber Newton polygon of φ equals the polygon definedas the lower convex hull of the points ( P ki =0 h i , P ki =0 ih i ) in R , where h i was defined in Section 2.) We assume17here exists a matrix A ( t ) such that the matrix B ( t ) of φ with respect to some orthonormal basis B satisfies B ( t ) = A σ a − ( t p a − ) · · · A σ ( t p ) A ( t ) (16)with A ( t ) satisfying (5). The latter condition means there exists b > A ( t ) all belongto L ( b ). Let φ i be the i -th slope unit root σ -module coming from the Hodge-Newton decomposition of φ . Wan’stheorem [30] tells us that the L -function of φ i is meromorphic: L ( φ i , A s / F q , T ) = Q ∞ i =1 (1 − α i T ) Q ∞ j =1 (1 − β j T ) , with α i , β j → i, j → ∞ . Theorem 2.6.
Assume the relative polytope (3) satisfies Γ ⊂ R ≥ . Suppose b ( p − ≤ . Then for all i and j , ord q α i and ord q β j ≥ b ( p − w (Γ) . A lower bound may also be given if b ( p − > .Proof. In the proof of Wan [30, Theorem 6.7] it was shown that there exists a sequence of matrices { B j ( t ) } withthe following properties. There exists matrices A j ( t ) for each j with entries in L ( b ), b independent of j , such that B j ( t ) = A σ a − j ( t p a − ) · · · A σj ( t p ) A ( t ) , with ord p ( A j ) ≥ j and lim j →∞ ord p ( A j ) = ∞ satisfying L ( φ i , A s / F q , T ) = ∞ Y j =1 L ( B j , A s / F q , T ) ± , where ± means the factor lies in either the numerator or the denominator. The result now follows by applyingpart (d) of Theorem 2.2 to each L -function in the product.When the unit root σ -module is of rank one, then we may allow the matrix A ( t ) to have possibly infinitedimension. This is useful in applications, as we will see in Section 4.3. Theorem 2.7.
Let ( M, φ ) be a possibly infinite rank nuclear σ -module, ordinary up to and including slope j .Suppose there exists b > and a matrix A ( t ) with entries in L ( b ) , Γ ⊂ R s ≥ , such that the matrix B ( t ) of φ withrespect to some orthonormal basis satisfies (16). Let φ j be the unit root σ -module coming from the j -th slope inthe Hodge-Newton decomposition of φ . Suppose φ j is of rank one. Then the same result as in Theorem 2.6 holdsfor the unit root L -function L ( φ j , A s / F q , T ) by the proof of [29, Theorem 8.5]. Notation.
Let V be a finite subset of Q n and define ∆( V ) as the convex closure of V ∪{ } in R n . Let Cone ( V ) be18he union of all rays from the origin through ∆( V ). Set M ( V ) := Z n ∩ Cone ( V ), a monoid. The monoid-algebra R ( V ) := F q [ M ( V )] may be filtered using a (polyhedral) weight function as follows. For each µ ∈ M ( V ), let w V ( µ )be the smallest non-negative rational number such that µ ∈ w V ( µ )∆( V ). Then w V ( M ( V )) ⊂ D ( V ) Z ≥ for somefixed positive integer D ( V ). The following properties of w V hold.(i) w V ( µ ) = 0 if and only if µ = 0.(ii) w V ( cµ ) = cw V ( µ ), if c ≥ µ ∈ M ( V ).(iii) w V ( µ + ν ) ≤ w V ( µ ) + w V ( ν ) for all µ, ν ∈ M ( V ).Furthermore, equality holds in (iii) if and only if µ and ν are cofacial with respect to the same closed face of∆( V ), i.e. the rays from 0 to µ and from 0 to ν intersect a common closed face of ∆( V ).The weight function w V imparts an increasing filtration to the ring R ( V ) defined byFil i R ( V ) := { ¯ g ∈ R ( V ) | w V ( µ ) ≤ i for all µ ∈ Supp(¯ g ) } for each i ∈ D ( V ) Z ≥ . The associated graded ring R ( V ) := gr R ( V ) has M ( V ) as a basis over F q and hasmultiplication rules x µ x ν = x µ + ν if µ and ν are cofacial with respect to a common closed face of ∆( V )0 otherwise . Toric family.
Let ¯ f ( x ) = P ¯ A ( µ ) x µ ∈ F q [ x ± , . . . , x ± n ]. Define Σ := Supp ( ¯ f ) := { µ ∈ Z n | ¯ A ( µ ) = 0 } . Using theconstruction above we define ∆( ¯ f ) := ∆(Σ), Cone ( ¯ f ) := Cone (Σ), M ( ¯ f ) := M (Σ), w := w Σ , R := R (Σ), and¯ R := ¯ R (Σ). We will assume throughout that1. dim ∆( ¯ f ) = n
2. ¯ f is nondegenerate with respect to ∆( ¯ f ). (Recall we write for a closed face σ of ∆( ¯ f ) not containing theorigin ¯ f σ = P µ ∈ σ ¯ A ( µ ) x µ . Then ¯ f is nondegenerate with respect to ∆( ¯ f ) if for every closed face σ ∈ ∆( ¯ f )not containing the origin { x ∂ ¯ f σ ∂x , . . . , x n ∂ ¯ f σ ∂x n } have no common solutions in ( F ∗ q ) n .)Let ¯ G ( x, t ) = ¯ f ( x ) + ¯ P ( x, t ) ∈ F q [ x ± , . . . , x ± n , t ± , . . . , t ± s ], with ¯ P ( x, t ) = P t γ ¯ P γ ( x ), where γ runs over a finitesubset T of Z s . We assume, in addition, that 0 ≤ w ( ν ) < ν ∈ S γ ∈ T Supp ¯ P γ ( x ). Note, condition (17) allows us to assume without loss that all monomials x µ inthe support of ¯ f have weight w ( µ ) = 1. (All monomials of weight less than 1 can be absorbed by the deformingLaurent polynomial ¯ P .) 19 elative polytope. Let U be a finite subset of Q s . As described above, any such set gives rise to data∆( U ), Cone ( U ), M ( U ), and R ( U ) in R s , and the weight function w U . For simplicity we consider subsets with Cone ( U ) = Cone ( T ), M ( U ) = M ( T ), with the property w U ( γ ) + w ( µ ) ≤ γ, µ ) ∈ Z s × Z n in Supp ( ¯ P ). Note that, since we are assuming (17), if U is chosen so that ∆( U ) issufficiently large then (18) will hold. We now define one which is minimal among all such and therefore givesoptimal estimates.Consider, for any U satisfying (18), the convex set∆( U ) × ∆( ¯ f ) ⊂ R n + s . Consider U := (cid:26)(cid:18) − w ( µ ) (cid:19) γ ∈ Q s | ( γ, µ ) ∈ Supp ( ¯ P ) (cid:27) , (19)and let Γ := ∆( U ). Then (18) implies for any such δ = (cid:16) − w ( µ ) (cid:17) γ ∈ U that w U ( δ ) ≤
1. As a consequenceΓ ⊂ ∆( U ) and w Γ ( γ ) ≥ w U ( γ ) for all γ ∈ M ( T ). On the other hand, by very definition, if δ = (cid:16) − w ( µ ) (cid:17) γ ∈ U then w Γ ( δ ) ≤ w Γ ( γ ) + w ( µ ) ≤ γ, µ ) ∈ Supp ( ¯ P ), and so (18) holds for U = U . Thus the choiceof Γ above makes w Γ optimal among weight functions w U satisfying (18). We call Γ the relative polytope of thefamily ¯ G ( x, t ).We will assume from now on that U = U and our weight function W on M ( T ) × M ( ¯ f ) is W ( γ, µ ) := w Γ ( γ ) + w ( µ ) . (20)Note that W is the weight function for the polyhedron Γ × ∆( ¯ f ) in R n + s . The Γ weight of a monomial ( γ, µ ) in M ( T ) × M ( ¯ f ) is w Γ ( γ ). We set S := F q [ M (Γ)] which is filtered using w Γ . Let ¯ S be the associated graded ring.Similarly, set M := F q [ M (Γ) × M ( ¯ f )], filtered using W defined in (20). M is an S -algebra. If ¯ M is the associatedgraded ring gr ( M ), then ¯ M is an ¯ S -algebra satisfying¯ S ( i ) ¯ M ( j ) ⊂ ¯ M ( i + j ) , (21)with multiplication obeying the rule ( t γ x µ )( t β x ν ) = t γ + β x µ + ν (22)if γ and β are cofacial with respect to a common closed face of Γ and µ and ν are cofacial with respect to acommon closed face of ∆( ¯ f ), else the multiplication equals 0. Rings of p -adic analytic functions. Let ζ p be a primitive p -th root of unity. Let Q q be the unramified20xtension of Q p of degree a := [ F q : F p ], and denote by Z q its ring of integers. Then Z q [ ζ p ] and Z p [ ζ p ] are the ringof integers of Q q ( ζ p ) and Q p ( ζ p ), respectively. Let π ∈ Q p ( ζ p ) be a zero of P ∞ j =0 t p j /p j having ord p ( π ) = 1 / ( p − Q q (containing Q q ( ζ p )). Say Q q (˜ π ) is a totallyramified extension of Q q containing Q q ( π ) with uniformizer ˜ π , a root of π . In this situation we denote by Z q [˜ π ](resp. Z p [˜ π ]) the ring of integers of Q q (˜ π ) (resp. Q p (˜ π )), and by σ the extension of the Frobenius generator of Gal ( Q q / Q p ) defined by σ (˜ π ) = ˜ π . Set O := X γ ∈ M (Γ) C ( γ ) t γ π w Γ ( γ ) | C ( γ ) ∈ Z q [˜ π ] , C ( γ ) → γ → ∞ . (We note again that the fractional powers of π are to be understood as integral powers of the uniformizer ˜ π .)Then O is a ring with a discrete valuation given by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X γ ∈ M (Γ) C ( γ ) t γ π w Γ ( γ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) := sup γ ∈ M (Γ) | C ( γ ) | . Note, it is not a valuation ring. In fact, the reduction map X γ ∈ M (Γ) C ( γ ) π w Γ ( γ ) t γ X γ ∈ M (Γ) ¯ C ( γ ) t γ identifies the rings O / ˜ π O ∼ −−→ ¯ S. Let C := X µ ∈ M ( ¯ f ) ξ ( µ ) π w ( µ ) x µ | ξ ( µ ) ∈ O , ξ ( µ ) → µ → ∞ , an O -algebra. For ξ ∈ C , write ξ = X µ ∈ M ( ¯ f ) ξ ( µ ) π w ( µ ) x µ = X ( γ,µ ) ∈ M (Γ) × M ( ¯ f ) ξ ( γ, µ ) π W ( γ,µ ) t γ x µ . The reduction map mod ˜ π , taking X ( γ,µ ) ∈ M (Γ) × M ( ¯ f ) ξ ( γ, µ ) π W ( γ,µ ) t γ x µ X ( γ,µ ) ∈ M (Γ) × M ( ¯ f ) ¯ ξ ( γ, µ ) t γ x µ identifies the ¯ S -algebras C / ˜ π C ∼ −−→ ¯ M . (23)21 he associated complex.
Define γ := 1, and for i ≥ γ i : = π − i X j =0 π p j p j = − π − ∞ X j = i +1 π p j p j . Note that ord p ( γ i ) = p i +1 − p − − ( i + 1) . Let G ( x, t ) be the lifting of ¯ G ( x, t ) using the Teichm¨uller units for all coefficients. Define H ( x, t ) := ∞ X i =0 γ i G σ i ( x p i , t p i )Notice that G σ i ( x p i , t p i ) has W -weight less than or equal to p i for all i ≥
1, and since x l ∂∂x l (cid:16) G σ i ( x p i , t p i ) (cid:17) = p i x l ∂G σ i ∂x l ! ( x p i , t p i ) , and ord p ( πγ i p i ) − p i p − p i − ≥ , we see that multiplication by πx l ∂H ( x,t ) ∂x l defines an endomorphism of C . Hence, we may define a complex of O -algebras Ω • ( C , ∇ G ) by Ω i ( C , ∇ G ) := M ≤ k < ··· Assume hypotheses 1 and 2 at the beginning of Section 3 above. Then the complex Ω • ( ¯ M , ∇ d ¯ G (1) ∧ ) is acyclic except in top dimension n . Furthermore, H n (Ω • ( ¯ M , ∇ d ¯ G (1) ∧ )) is a free ¯ S -algebra of rank equal to dim F q ( ¯ R/ P ni =1 x i ∂ ¯ f∂x i ¯ R ) = n ! vol (∆( ¯ f )) .If ¯ B is a basis of monomials such that the F q -vector space ¯ V spanned by ¯ B in ¯ R satisfies ¯ R = ¯ V ⊕ n X i =1 x i ∂ ¯ f∂x i ¯ R, then ¯ M = ¯ W ⊕ n X i =1 x i ∂ ¯ G (1) ∂x i ¯ M (24) where ¯ W is the free ¯ S -submodule of ¯ M generated by the same set of monomials ¯ B .Proof. For the first part of the theorem, we show that for every subset A of S := { , , . . . , n } the set { x i ∂ ¯ G (1) ∂x i } i ∈ A forms a regular sequence in ¯ M . If so, then H i (Ω • ( ¯ M , ∇ d ¯ G (1) ∧ )) = 0 for 0 ≤ i < n since the complex is in thiscase the Koszul complex on ¯ M defined by the elements { x i ∂ ¯ G (1) ∂x i } i ∈S . So, assume X i ∈ A x i ∂ ¯ G (1) ∂x i ¯ ξ i ( x, t ) = 0 (25)with ¯ ξ i ( x, t ) in ¯ M . Since ¯ M is graded and the { x i ∂ ¯ G (1) ∂x i } i ∈S are homogeneous of W -weight 1, it suffices to consider(25) in the case in which the { ¯ ξ i ( x, t ) } i ∈ A are homogeneous in ¯ M , say of W -weight k . We will prove that given2325), there exists then a skew-symmetric set { ¯ η il ( x, t ) } i,l ∈ A in ¯ M ( k − such that¯ ξ i ( x, t ) = X l ∈ A x l ∂ ¯ G (1) ∂x l ¯ η il ( x, t ) (26)for every i ∈ A . Now we consider for 0 ≤ ρ ≤ k representations of { ¯ ξ i } i ∈ A of the form¯ ξ i ( x, t ) = k X r ≥ ρ ¯ ξ ( r ) i ( x, t ) + X l ∈ A x l ∂ ¯ G (1) ∂x l ¯ γ il ( x, t ) (27)where ¯ ξ ( r ) i ( x, t ) = X w Γ ( γ )= r t γ ¯ ξ ( k − r ) iγ ( x )are terms of ¯ M ( k ) with w Γ -weight equal to r , and w -weight equal to k − r , and where { ¯ γ il ( x, t ) } i,l ∈ A is a skew-symmetric subset of ¯ M ( k − . Of course, we get such a representation in the case ρ = 0 by taking ¯ γ il = 0 for all i and l .Let D be the least common multiple of D (Γ) and D ( ¯ f ) so that W ( γ, µ ) = w Γ ( γ ) + w ( µ ) ∈ D Z ≥ . (28)We proceed by induction on ρ . More precisely, we show that given such a representation (27) for { ¯ ξ i ( x, t ) } i ∈ A with a given ρ = ρ we may find another such representation but with ρ ≥ ρ + D , so that in the end we producea representation with ρ > k hence of the form (26).It is convenient to write ¯ G (1) = ¯ f ( x ) + X We have1. Ω • ( ¯ M , ∇ ¯ G (1) ) is acyclic except in top dimension n .2. H n (Ω • ( ¯ M , ∇ ¯ G (1) )) = ¯ M / P nl =1 ¯ D l,t ¯ M is a free ¯ S -module of rank n ! vol ∆( ¯ f ) with basis ¯ B .3. We may write ¯ M = ¯ W ⊕ n X l =1 ¯ D l,t ¯ M . Furthermore, if t γ x µ ∈ ¯ M ( k ) then t γ x µ = X v i ∈ ¯ B a i,ν t ν v i + n X l =1 ¯ D l,t ¯ ξ l ( x, t ) (34) where W ( ν, v i ) ≤ k , w Γ ( ν ) ≥ w Γ ( γ ) and any term ( β, τ ) in the support of any ¯ ξ l has W ( β, τ ) ≤ k − and w Γ ( β ) ≥ w Γ ( γ ) . Finally, a slight modification of Theorem A1 [5, p.402], in which we drop the assumption that O is a discretevaluation ring, gives Theorem 3.3. Let O be a complete ring under a discrete valuation with uniformizer ˜ π and residue ring F = O / ˜ π O . Let C • = { → C ∂ −→ C ∂ −→ · · · C n → } be a length n cocomplex of flat, separated, complete O -modules with O -linear coboundary maps ∂ i . Let ¯ C • be thecocomplex obtained by reducing C • modulo ˜ π . Then1. For any i , H i ( ¯ C • ) = 0 implies H i ( C • ) = 0 .2. If H n ( ¯ C • ) is a free F -module of rank l , and H n − ( ¯ C • ) = 0 then H n ( C • ) is a finite free O -module of rank l . Using Theorem 3.3, we obtain the following corollary to Theorem 3.2: Theorem 3.4. The complex Ω • ( C , ∇ G ) is acyclic except in top dimension n , and H n (Ω • ( C , ∇ G )) is a free O -module of rank equal to n ! vol ∆( ¯ f ) . Furthermore, C = X v ∈ B O π w ( v ) v ⊕ n X l =1 D l,t C where B is a lifting of the monomials in ¯ B to characteristic zero and D l,t := x l ∂∂x l + πx l ∂H ( x, t ) ∂x l . .2 Frobenius It will be convenient in this section to denote F q by k . To motivate the development of the spaces C ( O ,q )defined below, formally define α := σ − ◦ πH ( x, t p ) ◦ ψ p ◦ exp πH ( x, t ) (35) α := 1exp πH ( x, t q ) ◦ ψ q ◦ exp πH ( x, t ) , where ψ p (cid:16)X A ( µ ) x µ (cid:17) = X A ( pµ ) x µ ψ q (cid:16)X A ( µ ) x µ (cid:17) = X A ( qµ ) x µ , and σ ∈ Gal ( Q q ( ζ p ) / Q p ( ζ p )) is the Frobenius generator which we extend to Gal ( K/K ) taking σ (˜ π ) = ˜ π . Sinceformally, D l,t = 1exp πH ( x, t ) ◦ x l ∂∂x l ◦ exp πH ( x, t )the following commutation laws will hold for l = 1 , , . . . , n , qD l,t q ◦ α = α ◦ D l,t and pD l,t p ◦ α = α ◦ D l,t . (36)Since the differential operators D l,t commute with α by changing t to either t p or t q , in order to proceed, weneed to introduce some new spaces. In the following, q = p a is an arbitrary power of p (including the case when a = 0), so we can handle the cases of t q , t p , and t , at the same time. Define O ,q := X γ ∈ M (Γ) A ( γ ) t γ π w q Γ ( γ ) | A ( γ ) ∈ Z q [ π ] , A ( γ ) → γ → ∞ . (37)This ring is the same as O except using a weight function defined by the dilation q Γ (that is, w q Γ ( γ ) = w Γ ( γ ) /q ).We note that here O , = O . A discrete valuation may be defined as follows. If ξ = P γ ∈ M (Γ) A ( γ ) π w q Γ ( γ ) t γ then | ξ | := sup γ ∈ M (Γ) | A ( γ ) | . We may also define the space C ( O ,q ) := X µ ∈ M ( ¯ f ) ξ µ x µ π w ( µ ) | ξ µ ∈ O ,q , ξ µ → µ → ∞ . (38)28e will sometimes write C as C ( O ). For η = P µ ∈ M ( ¯ f ) ξ µ π w ( µ ) x µ , we set | η | = sup µ ∈ M ( ¯ f ) | ξ µ | . The reduction map X A ( γ ) t γ π w q Γ ( γ ) X A ( γ ) t γ takes O ,q to the graded ring ¯ S q := gr k [ M (Γ)], where k [ M (Γ)] has been graded using w q Γ . In this case, ¯ S q isidentical to ¯ S defined earlier but regraded so that ¯ S ( i/q ) q = ¯ S ( i ) .Replacing the weight function (28) with W q ( γ, µ ) := w q Γ ( γ ) + w ( µ ) ∈ D ( q ) Z ≥ , (39)where D ( q ) is the least common multiple of D and q , the proofs of Theorems 3.1, 3.2, and 3.4 now may bemodified slightly so that analogous versions hold for these spaces, using M q := k [ M ( q Γ) × M ( ¯ f )] filtered by W q and ¯ M q := gr M q . Of course, since M (Γ) = M ( q Γ), we see that ¯ M q is just ¯ M regarded with ¯ M ( i/q ) q = ¯ M ( i ) . Wegive explicitly the statement of our analogue of Theorem 3.4: Theorem 3.5. Let D l,t q := x l ∂∂x l + πx l ∂H ( x,t q ) ∂x l . Let Ω • ( C ( O ,q ) , ∇ ¯ G ( x,t q ) ) be the complex with Ω i ( C ( O ,q ) , ∇ ¯ G ( x,t q ) ) = M ≤ j < ··· 1) be a rational number. Define R ( b ; c ) := X γ ∈ M (Γ) A ( γ ) t γ | A ( γ ) ∈ Q q ( π ) , ord p A ( γ ) ≥ bw Γ ( γ ) + c R ( b ) := [ c ∈ R R ( b ; c ) . 29e define a valuation on R ( b ) as follows. If ξ = P γ ∈ M (Γ) A ( γ ) t γ ∈ R ( b ), then ord p ξ = inf γ ∈ M (Γ) { ord p A ( γ ) − w Γ ( γ ) b } = sup { c ∈ R | ξ ∈ R ( b ; c ) } . Note that R ( b ; c ) R ( b ; c ′ ) ⊂ R ( b ; c + c ′ )since X γ ∈ M (Γ) A ( γ ) t γ X ˜ γ ∈ M (Γ) A ′ (˜ γ ) t ˜ γ = X β ∈ M (Γ) X γ +˜ γ = β A ( γ ) A ′ (˜ γ ) t β . If β is fixed and ( γ, ˜ γ ) runs through pairs in M (Γ) such that γ + ˜ γ = β , then sup( | γ | , | ˜ γ | ) → ∞ so thatinf( | A ( γ ) | , | A ′ (˜ γ ) | ) → P γ +˜ γ = β A ( γ ) A ′ (˜ γ ) converges. Thatinf ord p X γ +˜ γ = β A ( γ ) A ′ (˜ γ ) − w Γ ( β ) b ≥ c + c ′ is clear.For the moment, let R be any ring with a p -adic valuation. For b ∈ R ≥ and c ∈ R , set L ( b, c ; R ) := X µ ∈ M ( f ) ξ µ x µ | ξ µ ∈ R and ord p ( ξ µ ) ≥ bw ( µ ) + c . In particular, if R = R ( b ′ ; c ′ ) with c ′ ≥ 0, we may write ξ µ = P γ ∈ M (Γ) A γ,µ t γ with ord p ( A γ,µ ) ≥ b ′ w Γ ( γ ) + c ′ forall γ , so that L ( b, c ; R ) = X ( γ,µ ) ∈ M (Γ) × M ( f ) A γ,µ t γ x µ | ord p ( A γ,µ ) ≥ b ′ w Γ ( γ ) + bw ( µ ) + c . This motivates our definition of the spaces K ( b ′ , b ; c ) below. Let 0 < b ′ , b ≤ p/ ( p − 1) be rational numbers, andlet c ∈ R . Define K ( b ′ , b ; c ) := X ( γ,µ ) ∈ M (Γ) × M ( ¯ f ) A γ,µ t γ x µ | A γ,µ ∈ Q q ( π ) , ord p A γ,µ ≥ b ′ w Γ ( γ ) + bw ( µ ) + c K ( b ′ , b ) := [ c ∈ R K ( b ′ , b ; c ) . We consider now the Frobenius maps α and α on C and on K ( b ′ , b ). Recall the Artin-Hasse series E ( t ) =exp (cid:16)P ∞ j =0 t p j /p j (cid:17) together with π , a zero of P ∞ j =0 t p j /p j satisfying ord p π = 1 / ( p − θ ( t ) := E ( πt ) = ∞ X j =0 θ j t j . Its coefficients satisfy ord p θ j ≥ j/ ( p − G ( x, t ) = X ¯ A ( γ, µ ) t γ x µ ∈ F q [ x ± , . . . , x ± n , t ± , . . . , t ± s ] , we let G ( x, t ) := X A ( γ, µ ) t γ x µ ∈ Z q [ x ± , . . . , x ± n , t ± , . . . , t ± s ]be the lifting of ¯ G by Teichm¨uller units. Set F ( x, t ) := Y ( γ,µ ) ∈ Supp ( ¯ G ) θ ( A ( γ, µ ) t γ x µ ) ∈ K ( 1 p − , p − ⊂ K ( b ′ /p, b/p ; 0) (40)and F a ( x, t ) := a − Y i =0 F σ i ( x p i , t p i ) ∈ K ( pq ( p − , pq ( p − 1) ; 0) ⊂ K ( b ′ /q, b/q ; 0) , (41)where σ is the Frobenius generator of Gal ( Q q ( π ) / Q p ( π )) acting on the coefficients of F . Note that C ⊂ K ( 1 p − , p − ⊂ K ( b ′ /p, b/p )so that multiplication by F takes C , as well as K ( b ′ /p, b/p ), into K ( b ′ /p, b/p ), and multiplication by F a takesthese two spaces into K ( b ′ /q, b/q ). It is easy to see ψ p ( K ( b ′ , b ; c )) ⊂ K ( b ′ , pb ; c ) ψ q ( K ( b ′ , b ; c )) ⊂ K ( b ′ , qb ; c ) . Finally, we note that for b ′ and b > / ( p − K ( b ′ /q, b ; 0) ⊂ C ( O ,q ). Therefore, since ψ p acts on the x -variables, α := σ − ◦ ψ p ◦ F ( x, t )maps σ − -semilinearly C ( O ) into C ( O ,p ), and it maps σ − -semilinearly K ( b ′ , b ; c ) into K ( b ′ /p, b ; c ). Similarly,if we define α := ψ q ◦ F a ( x, t ) , then α maps C ( O ) into C ( O ,q ) linearly over Z q [ π ], as well as K ( b ′ , b ; c ) into K ( b ′ /q, b ; c ).31e may use α and α to define chain maps as follows. Let F rob i := M ≤ j < ··· Let < b ≤ p/ ( p − and c ∈ R . Then the following equalities hold for any q a power of p .1. K ( b/q, b ; c ) = W ( b/q, b ; c ) + P ni =1 πG (1) i ( x, t q ) K ( b/q, b ; c + e ) K ( b/q, b ; c ) = W ( b/q, b ; c ) + P ni =1 πG i ( x, t q ) K ( b/q, b ; c + e ) where W ( b/q, b ; c ) := X γ ∈ M (Γ) ,v ∈ B A ( γ, v ) t γ v | A ( γ, v ) ∈ Q q (˜ π ) , ord p A ( γ, v ) ≥ bW q ( γ, v ) + c In particular, W ( b/q, b ; 0) = K ( b/q, b ; 0) ∩ M v ∈ B O ,q π w ( v ) v. Proof. The right side of these equalities is clearly contained in the left side of the corresponding equality. We firstconcentrate on the first equality. Let ξ = P ( γ,µ ) ∈ M (Γ) × M ( ¯ f ) A ( γ, µ ) t γ x µ ∈ K ( b/q, b ; c ). Let ( γ, µ ) ∈ M (Γ) × M ( ¯ f )with W q ( γ, µ ) = i ∈ D ( q ) Z ≥ , as defined in (39). We know from (24) that we may write in characteristic pt γ x µ = X v ∈ ¯ B,W q ( β,v )= i,w q Γ ( β ) ≥ w q Γ ( γ ) C (( γ, µ ) , ( β, v )) t β v + n X j =1 ¯ G (1) j ( x, t q )¯ η j where C (( γ, µ ) , ( β, v )) ∈ F q and ¯ η j ∈ ¯ M ( i − q . 32e lift the coefficients C (( γ, µ ) , ( β, v )) and those of the polynomial ¯ η j to characteristic zero using (23) in theform C ( O ,q ) / ˜ π C ( O ,q ) ∼ = ¯ M q . Then, for each ( γ, µ ), and e := p − · D ( q ) , we have ord p π W q ( γ,µ ) t γ x µ − X v ∈ B C (( γ, µ ) , ( β, v )) π W q ( β,v ) t β v ! − n X j =1 G (1) j ( x, t q ) η j ( γ, µ ) ≥ e . Since ord p (cid:16) A ( γ,µ ) π Wq ( γ,µ ) (cid:17) ≥ 0, multiplication by this gives a similar result: ord p A ( γ, µ ) t γ x µ − X v ∈ B A ( γ, µ ) π W q ( γ,µ ) C (( γ, µ ) , ( β, v )) π W q ( β,v ) t β v ! − n X j =1 G (1) j ( x, t q ) A ( γ, µ ) π W q ( γ,µ ) η j ( γ, µ ) ≥ e . Now, ω ( γ, µ ) := X v ∈ B A ( γ, µ ) π W q ( γ,µ ) C (( γ, µ ) , ( β, v )) π W q ( β,v ) t β v ∈ W ( bq , b ; c ) , and if we set ζ j := π − A ( γ,µ ) π Wq ( γ,µ ) η j ( γ, µ ) then ζ j ∈ K ( bq , b ; c + e ). Hence, A ( γ, µ ) t γ x µ − ω ( γ, µ ) + n X j =1 πG (1) j ( x, t q ) ζ j ( γ, µ ) ∈ K ( bq , b ; c + e ) . We obtain then ξ = ω (0) + P nj =1 πG (1) j ( x, t q ) ζ (0) j + ξ (1) where ω (0) = P ( γ,µ ) ∈ M (Γ) × M ( ¯ f ) ω ( γ, µ ), ζ (0) j = P ( γ,µ ) ∈ M (Γ) × M ( ¯ f ) ζ j ( γ, µ ), and ξ (1) ∈ K ( b/q, b ; c + e ). Iterating the argument above, we obtain for every N ∈ Z ≥ , ω ( N ) ∈ W ( b/q, b ; c + N e ), ζ ( N ) j ∈ K ( b/q, b ; c + e + N e ) for 1 ≤ j ≤ n and ξ ( N ) ∈ K ( b/q, b ; c + N e )with ξ ( N ) = ω ( N ) + n X j =1 πG (1) j ( x, t q ) ζ ( N ) j + ξ ( N +1) so ξ = ξ ( N +1) + N X l =0 ω ( l ) + n X j =1 πG (1) j ( x, t q ) N X l =0 ζ ( l ) j ! . Letting N → ∞ , we obtain that the left hand side of the first equality in Theorem 3.6 is contained in the righthand side.We now prove the second part of the theorem. Using the first equality of the theorem, we may write each ξ ∈ K ( b/q, b ; c ) as ξ = ω + n X j =1 πG (1) j ( x, t q ) ζ j with ω ∈ W ( b/q, b ; c ) and ζ j ∈ K ( b/q, b ; c + e ). But then ξ = ω + n X j =1 πG j ( x, t q ) ζ j − n X j =1 πg j ( x, t q ) ζ j and P nj =1 πg j ( x, t q ) ζ j ∈ K ( b/q, b ; c + D ( q ) ). Iterating this finishes the proof.33ecall, we have defined H ( x, t ) = ∞ X j =0 γ j G σ j ( x p j , t p j )with ord p γ j = p j +1 p − − ( j + 1) for j ≥ 0. Recall also that D l,t q := x l ∂∂x l + πH l ( x, t q ) , where H l ( x, t q ) := x l ∂∂x l H ( x, t q ). Theorem 3.7. For b a rational number satisfying / ( p − < b ≤ p/ ( p − , c ∈ R , and q a power of p , we have K ( b/q, b ; c ) = W ( b/q, b ; c ) + n X l =1 πH l ( x, t q ) K ( b/q, b ; c + e ) . (48) Proof. It follows from the bound b ≤ p/ ( p − p -adic order of γ j above, that πH l ( x, t q ) ∈ K ( b/q, b ; c + e ). This gives us that the right hand side of (48) is contained in the left hand side. To establish thereverse-inclusion, note that G σ j l ( x p j , t qp j ) = G l ( x, t q ) p j + ph l,j ( x, t q )where h l,j ( x, t ) has integral coefficients and all monomials ( β, ν ) in h l,j ( x, t q ) have W q -weight less than or equalto p j . Thus, we may write πH l ( x, t q ) = πG l ( x, t q ) Q l ( x, t q ) + K l ( x, t q )where Q l ( x, t q ) := ∞ X m =0 γ m p m G l ( x, t q ) p m − K l ( x, t q ) := ∞ X m =1 γ m p m +1 h l,m ( x, t q ) . Consequently, Q l ( x, t q ) , Q l ( x, t q ) − , K l ( x, t q ) ∈ K ( pq ( p − , pp − . (49)So, if 1 / ( p − < b ≤ p/ ( p − 1) and ξ ∈ K ( b/q, b ; c ) then there exists ω ∈ W ( b/q, b ; c ) and ζ l ∈ K ( b/q, b ; c + e ) byTheorem 3.6 such that ξ = ω + n X l =1 πG l ( x, t q ) ζ l = ω + n X l =1 πG l ( x, t q ) Q l ( x, t q ) Q l ( x, t q ) − ζ l = ω + n X l =1 πH l ( x, t q ) Q l ( x, t q ) − ζ l − n X l =1 K l ( x, t q ) Q l ( x, t q ) − ζ l Q l ( x, t q ) − ζ l and P nl =1 K l ( x, t q ) Q l ( x, t q ) − ζ l belonging to K ( b/q, b ; c + e ). Since b > / ( p − e > Theorem 3.8. For b a rational number satisfying / ( p − < b ≤ p/ ( p − , c ∈ R , q a power of p , we have K ( b/q, b ; c ) = W ( b/q, b ; c ) + n X l =1 D l,t q K ( b/q, b ; c + e ) . Proof. Again, the right hand side is contained in the left side. For the reverse inclusion, let ξ ∈ K ( b/q, b ; c ). Weknow ξ = ω + n X l =1 πH l ( x, t q ) ζ l where ω ∈ W ( b/q, b ; c ) and ζ l ∈ K ( b/q, b ; c + e ) by Theorem 3.7. But then ξ = ω + n X l =1 D l,t q ζ l − n X l =1 x l ∂ζ l ∂x l . Since P x l ∂ζ l ∂x l ∈ K ( b/q, b ; c + e ), the theorem follows by a similar recursive argument. Theorem 3.9. For b a rational number satisfying / ( p − < b ≤ p/ ( p − , we have K ( b/q, b ) = W ( b/q, b ) ⊕ n X l =1 D l,t q K ( b/q, b ) . Proof. By the previous theorem, it only remains to show the sum on the right is direct. Suppose on the contrary ω = P nl =1 D l,t q ζ l . Without loss of generality, we may assume both ω ∈ K ( b/q, b ; 0) and each ζ l ∈ K ( b/q, b ; e ) ⊂ K ( b/q, b ; 0). Since b > / ( p − K ( b/q, b ; 0) ⊂ C ( O ,q ) , so that ω = P nl =1 D l,t ζ l implies ω = 0 and ζ l = 0 for every l by Theorem 3.5.We are now able to provide the following estimates for the entries of the Frobenius. First, note that α ( x µ ) ∈ K ( b/p, b ; − ( b/p ) w ( µ )) , and so by Theorem 3.8, we may write α ( x µ ) = X v ∈ B A µ,v v mod n X l =1 D l,t p K ( b/p, b )with A µ,v ∈ L ( b/p ; ( b/p )( pw ( v ) − w ( µ ))) . (50)35 .3 L -functions of the toric family In this section, we will apply Theorem 2.2 to prove Theorem 1.1.Let λ ∈ F ∗ sq , with deg ( λ ) := [ F q ( λ ) : F q ]. Using Dwork’s splitting function, define an additive characterΘ : F q → Q p by Θ := θ (1) T r F q/ F p ( · ) , and Θ λ := Θ ◦ T r F q ( λ ) / F q . Define S r ( λ ) := X x ∈ F ∗ nqrdeg ( λ ) Θ λ ◦ T r F qrdeg ( λ ) / F q ( λ ) ¯ G ( x, λ )and its L -function L ( ¯ G λ , T ) := L ( ¯ G λ , Θ , G nm / F q ( λ ) , T ) := exp ∞ X r =1 S r ( λ ) T r r ! . Let ˆ λ be the Teichm¨uller representative of λ . Let O , ˆ λ = Z q [˜ π, ˆ λ ]. There is an obvious ring map, which wecall the specialization map at ˆ λ , from O to O , ˆ λ induced by the map sending t ˆ λ . Similarly, let C , ˆ λ be the O , ˆ λ -module obtained by specializing the space C at t = ˆ λ . Let α ˆ λ := α deg ( λ ) | t =ˆ λ , and define F rob i ˆ λ as in (42)but with α replaced by α ˆ λ . Then using Dwork’s trace formula, we obtain S r ( λ ) = ( q rdeg ( λ ) − n T r ( α ˆ λ | C , ˆ λ )= n X i =0 ( − i T r ( H i ( F rob i ˆ λ ) r | H i ( C , ˆ λ , ∇ ¯ G ( x, ˆ λ ) )) . Since cohomology is acyclic by [5] except in top dimension n , we have S r ( λ ) = ( − n T r ( H n ( F rob n ˆ λ ) r | H n ( C , ˆ λ , ∇ ¯ G ( x, ˆ λ ) )) . In other words, writing ¯ α ˆ λ for H n ( F rob n ˆ λ ), we have L ( ¯ G λ , T ) ( − n +1 = det (1 − ¯ α ˆ λ T )= (1 − π ( λ ) T ) · · · (1 − π N ( λ ) T ) , where N = n ! vol (∆ ∞ ( ¯ f )). In [5] it is proved for each such λ that the Newton polygon of L ( ¯ G λ , T ) ( − n +1 liesover the Newton polygon (using ord ˆ λ ) of Y β ∈B (1 − ( q deg ( λ ) ) w ( β ) T ) . (51)For each λ ∈ ( F × q ) s , set A ( λ ) := { π i ( λ ) } Ni =1 the collection of eigenvalues of ¯ α ˆ λ . Let L be a linear algebraoperation. Let LA ( λ ) be the set of eigenvalues of L ¯ α ˆ λ . Define L ( LA , G sm / F q , T ) := Y λ ∈| G sm / F q | Y τ ( λ ) ∈LA ( λ ) (1 − τ ( λ ) T deg ( λ ) ) − . 36o aid the reader, we will consider a running example throughout this section; if L is the operation of the k -thsymmetric power tensor the l -th exterior power, then LA ( λ ) = Sym k A ( λ ) ⊗ ∧ l A ( λ )= { π ( λ ) i · · · π N ( λ ) i N π j ( λ ) · · · π j l ( λ ) | i + · · · + i N = k, ≤ j < · · · < j l ≤ N } . Note, the cardinality of LA ( λ ) is independent of λ ; let L N denote this number.Let B = { x µ , . . . , x µ N } be a basis for H n := H n (Ω • ( C , ∇ G )). For q a power of the prime p (perhaps q = p )define L H nq := L H n (Ω • ( C ( O ,q ) , ∇ G ( x,t q ) )) . This is a free O ,q -module with basis LB = { e i } i ∈ I , for some index set I . Note, LB is a basis of L H nq for everyprime power q . In our example L H n = Sym k H n ⊗ ∧ l H n , elements in the basis LB take the form e i = ( x µ ) i · · · ( x µ N ) i N ⊗ ( x µ j ∧ · · · ∧ x µ jl )where i + · · · + i N = k and 1 ≤ j < · · · < j l ≤ N. We extend the Frobenius map to this space by defining L ¯ α := L H n ( F rob n ) : L H n → L H nq . Let B ( t ) be the matrix of L ¯ α with respect to the basis LB . Let A ( t ) be the matrix of ¯ α with respect to the basis B , then the matrix of ¯ α ˆ λ is A ˆ λ := A (ˆ λ q deg ( λ ) − ) · · · A (ˆ λ q ) A (ˆ λ ). Similarly the matrix of L ¯ α ˆ λ using the basis LB is L A ˆ λ . We have B (ˆ λ q deg ( λ ) − ) · · · B (ˆ λ q ) B (ˆ λ ) = L A ˆ λ . Since the set of eigenvalues of L A ˆ λ is LA ( λ ), we have det (1 − B (ˆ λ q deg ( λ ) − ) · · · B (ˆ λ q ) B (ˆ λ ) T deg ( λ ) ) = Y τ (ˆ λ ) ∈LA ( λ ) (1 − τ (ˆ λ ) T deg ( λ ) ) . Consequently, L ( LA , G sm / F q , T ) := Y λ ∈| G sm / F q | Y τ (ˆ λ ) ∈LA ( λ ) (1 − τ (ˆ λ ) T deg ( λ ) ) − = Y λ ∈| G sm / F q | det (1 − B (ˆ λ q deg ( λ ) ) · · · B (ˆ λ q ) B (ˆ λ ) T deg ( λ ) ) − =: L ( B, G sm , T ) 37s in (4). Proposition 3.10. L ( LA , G sm / F q , T ) is a rational function over Q ( ζ p ) .Proof. Writing L ( LA , G sm / F q , T ) = exp (cid:0)P ∞ m =1 N m T m m (cid:1) , we have N m = X λ ∈ ( F ∗ qm ) s X τ ( λ ) ∈LA ( λ ) deg ( λ ) τ ( λ ) m/deg ( λ ) . Since ¯ G λ is nondegenerate for each λ , by [5] and [12] each eigenvalue has Archimedean weight at most n . Em-bedding into C , this means | π i ( λ ) | C ≤ q n deg ( λ ) / . Since each τ ( λ ) is a product of |L| eigenvalues π i ( λ ), with eachfactor having weight at most n , we have | N m | C ≤ m L N · q |L| nm/ . Thus, L ( LA , G sm / F q , T ) has a positive radiusof convergence over C .Next, since L ( ¯ G λ , T ) is a rational function over Z [ ζ p ], the polynomial Q τ ( λ ) ∈LA ( λ ) (1 − τ ( λ ) T deg ( λ ) ) hascoefficients in Z [ ζ p ]. It follows that the coefficients of the power series expansion of L ( LA , G sm / F q , T ) lie in afixed number field Q ( ζ p ). Since L ( LA , G sm / F q , T ) is both p -adic meromorphic, as in (6) above, and converges ona disc of positive radius over C , rationality follows from the Borel-Dwork theorem [17, Section 4].Define a weight on each basis vector in LB = { e i } i ∈ I as follows. Let w ( i ) := N X i =1 m i w ( µ i ) (52)where x µ i appears m i -times in the basis element e i . For example, continuing our running example, if e i = ( x µ ) i · · · ( x µ N ) i N ⊗ ( x µ j ∧ · · · ∧ x µ jl )then w ( i ) = i w ( µ ) + · · · + i N w ( µ N ) + w ( µ j ) + · · · + w ( µ j l ) . Proposition 3.11. There exists a matrix A = ( A i , j ) i , j ∈ I with entries in L ( p − ) such that B ( t ) = A σ a − ( t p a − ) · · · A σ ( t p ) A ( t ) and A i , j ∈ L (1 / ( p − p − pw ( j ) − w ( i ))) . (53) Proof. From (35), writing α ( t ) for α and α ( t ) for α , and writing ψ x,p for ψ p acting only on the x variables, wehave α ( t ) = ψ ax,p ◦ F a ( x, t )= ψ ax,p ◦ F σ a − ( x p a − , t p a − ) · · · F σ ( x p , t p ) F ( x, t )= (cid:16) σ − ◦ ψ x,p ◦ F ( x, t p a − ) (cid:17) ◦ · · · ◦ (cid:0) σ − ◦ ψ x,p ◦ F ( x, t p ) (cid:1) ◦ (cid:0) σ − ◦ ψ x,p ◦ F ( x, t ) (cid:1) = α ( t p a − ) ◦ · · · ◦ α ( t p ) ◦ α ( t ) . (54)38enote by ¯ α ( t ) and ¯ α ( t ) the maps H n ( F rob n ) and H n ( F rob n ), respectively. Then (54) shows¯ α ( t ) = ¯ α ( t p a − ) ◦ · · · ◦ ¯ α ( t p ) ◦ ¯ α ( t ) . Consequently, L ¯ α ( t ) = L ¯ α ( t p a − ) ◦ · · · ◦ L ¯ α ( t p ) ◦ L ¯ α ( t ) . (55)Let A ( t ) be the matrix of L ¯ α with respect to the basis LB . Then the matrix version of (55) is (53).We now proceed to the estimates on A ( t ). We extend the weight function (39) as follows W q ( γ, i ) := w q Γ ( γ ) + w ( i ) ∈ D ( q ) Z ≥ for γ ∈ M (Γ) and i ∈ I . Define the spaces, for q a power of p (perhaps with q = p ) and c ∈ R , LW ( b/q, b ; c ) := X γ ∈ M (Γ) , i ∈ I A ( γ, i ) t γ e i | A ( γ, i ) ∈ Q q (˜ π ) , ord p ( A ( γ, i )) ≥ bW q ( γ, i ) + c L W ( b/q, b ) := [ c ∈ R L W ( b/q, b ; c ) . Then, for any rational b satisfying 1 / ( p − < b ≤ p/ ( p − L ¯ α : L W ( b/p, b/p ; 0) → L W ( b/p, b ; 0)and A i , j ∈ L ( bp ; bp ( pw ( j ) − w ( i ))) . Setting b = p/ ( p − 1) to get the best possible p -adic estimates, we have A i , j ∈ L ( 1 p − p − pw ( j ) − w ( i ))) . Proof of Theorem 1.1. If we modify the basis LB by the following normalization, ˜ e i := π w ( i ) e i for each i ∈ I , thenthe matrix of L ¯ α with respect to this basis takes the form ˜ A ( t ) = ( A i , j π w ( i ) − w ( j ) ), with entries satisfying˜ A i , j := A i , j π w ( i ) − w ( j ) ∈ L (1 / ( p − w ( j )) . Then if ˜ B ( t ) is the matrix of L ¯ α with respect to this basis, we have˜ B ( t ) = ˜ A σ a − ( t p a − ) · · · ˜ A σ ( t p ) ˜ A ( t ) . 39e employ Proposition 3.11 with the basic data b = p − , ramification e = p − { s ( j ) = ( p − w ( j ) } j ∈ I . Since L ( B, G sm / F q , T ) = L ( ˜ B, G sm / F q , T ), we may apply Theorem 2.2 to obtain Theorem 1.1 and Theorem 1.2 (a) .Parts (b) and (c) of Theorem 1.2 follow immediately; see [1, p.557]. What remains is the determination of k (andconsequently ρ ) in the statement of Theorem 2.2 (c) . We are indebted to Nick Katz for the following argument,which will show that k = s + n |L| .Let L Θ be the ℓ -adic sheaf on A / F q corresponding to the character Θ of F q . Viewing ¯ G as a map ¯ G : G nm × G sm → A defined over F q , let L Θ( ¯ G ) be the pullback of L Θ to G nm × G sm . Let π : G nm × G sm → G sm bethe projection onto the second factor. It follows that R i π L Θ( ¯ G ) = 0 for every i = n (since by [5] and [12] everystalk is zero). Further, these same references show that R n π L Θ( ¯ G ) has constant rank equal to n ! vol ∆ ∞ ( ¯ f ).The sheaf R n π L Θ( ¯ G ) is, in Katz’s terminology, of perverse origin [24, Corollary 6]. Since this sheaf has constantrank, it follows from [24, Proposition 11] that R n π L Θ( ¯ G ) is lisse. It makes sense therefore to apply a linearalgebra operation such as L to this sheaf. We view L as a quotient of some r -fold tensor product; the minimumsuch r we denote by |L| . Then L R n π L Θ( ¯ G ) is mixed with weights ≤ |L| n . The eigenvalues of Frobenius actingon H ic, ´et ( G sm / F q , L R n π L Θ( ¯ G ) ) have weights ≤ i + |L| n for any i in the range 0 ≤ i ≤ s . All eigenvalues arealgebraic integers so that the weight of any eigenvalue γ bounds the valuation:0 ≤ ord q γ ≤ weight( γ ) . Thus for i ≤ s , the p -divisibility of any eigenvalue of Frobenius acting on H ic, ´et ( G sm / F q , L R n π L Θ( ¯ G ) ) satisfies ord q γ ≤ s + |L| n. We now show this inequality holds as well for eigenvalues of Frobenius on H ic, ´et ( G sm / F q , L R n π L Θ( ¯ G ) ) with i inthe upper range s < i ≤ s . For these, we invoke the work of Deligne [11, Corollary 3.3.3] which applies since L R n π L Θ( ¯ G ) is an integral sheaf. For γ an eigenvalue of Frobenius acting on H ic, ´et ( G sm / F q , L R n π L Θ( ¯ G ) ) with s < i ≤ s , Deligne’s result implies that γ/q i − s is an algebraic integer and pure of some weight. Thus, as above, ord q ( γ/q i − s ) ≤ weight( γ/q i − s ) ., so that ord q ( γ ) − ( i − s ) ≤ weight( γ ) − i − s ) ≤ i + |L| n − i − s ) . Thus, ord q γ ≤ s + |L| n . 40 Other families We now state two related theorems which follow from the work and results above in a well-known manner (see[5] and [26]). Let ¯ f ( x ) ∈ F q [ x ± , . . . , x ± r , x r +1 , . . . , x n ] and set S := { x , . . . , x n } , S := { x , . . . , x r } , S := { x r +1 , . . . , x n } . We will continue to assume the hypotheses and , dim ∆ ∞ ( ¯ f ) = n and ¯ f nondegenerate withrespect to ∆ ∞ ( ¯ f ), on ¯ f from the beginning of Section 3. We say ¯ f is convenient with respect to S if for all subsets A ⊂ S , dim ∆ ∞ ( ¯ f A ) = n − | A | , where ¯ f A is the Laurent polynomial in n − | A | variables obtained by setting eachvariable x i with i ∈ A equal to zero. We will also use the notation ¯ M ( A ) (and C ( A )0 respectively) for the elementsin ¯ M (and C ) which have support in the set of monomials x µ = x µ · · · x µ n n in M ( ¯ f ) satisfying µ i ≥ i ∈ A . Then ¯ M ( A ) and C ( A )0 are ideals in ¯ M and C respectively. DefineΩ i ( C , S , ∇ ¯ G ) := M A = { ≤ j < ··· For i = n , H i (Ω • ) = 0 for all three complexes Ω • ( C , S , ∇ ¯ G ) , Ω • ( ¯ M , S , ∇ ¯ G ) , and Ω • ( ¯ M , S , ∇ d ¯ G (1) ∧ ) defined above. Furthermore H n (Ω • ( C , S , ∇ ¯ G )) is a free O -module of rank υ S ( ¯ f ) = X A ⊂S ( − | A | ( n − | A | )! vol A (∆ ∞ ( ¯ f A )) , where vol A (∆ ∞ ( ¯ f A )) is the volume with respect to Lebesgue measure on R nA := { x = ( x , . . . , x n ) ∈ R n | x i =0 if i ∈ A } .Furthermore, we set ¯ B ( S ) = S i ∈ D Z ≥ ¯ B ( S ,i ) where ¯ B ( S ,i ) is a subset of the set of monomials in ¯ R ( S ) ofweight i such that ¯ V ( S ,i ) , the F q -space spanned by ¯ B ( S ,i ) , satisfies ¯ R ( S ,i ) = ¯ V ( S ,i ) ⊕ n X l =1 x l ∂ ¯ f∂x l ¯ R ( S −{ l } ,i − . (Here S − { l } equals S if l 6∈ S .) Then C ( S )0 = X v ∈ ¯ B ( S O π w Γ ( v ) v ⊕ n X l =1 D l,t C ( S −{ l } )0 . (57)41s we did in the previous section, let λ ∈ F ∗ sq , with deg ( λ ) := [ F q ( λ ) : F q ]. Define the exponential sums S l ( λ ) := X x ∈ F ∗ rqldeg ( λ ) × F n − rqldeg ( λ ) Θ λ ◦ T r F qldeg ( λ ) / F q ¯ G ( x, λ )and the associated L -function on A n − r × G rm / F q ( λ ): L ( ¯ G λ , A n − r × G rm / F q ( λ ) , T ) := exp ∞ X r =1 S r ( λ ) T r r ! . Let ˆ λ be the Teichm¨uller representative of λ . Let O ,λ be the ring Z q [ π, ˆ λ ], and let C ,λ be the O ,λ -moduleobtained by specializing the space C at t = ˆ λ . Let α ˆ λ := α deg ( λ ) | t =ˆ λ , and define F rob i ˆ λ as in (42) but with α replaced by α ˆ λ . Then F rob • ˆ λ is a chain map on Ω • ( C ,λ , S , ∇ ¯ G ( x,λ ) ). Write ¯ α ˆ λ for H n ( F rob n ˆ λ ) acting on H n (Ω • ( C , ˆ λ , S , ∇ ¯ G ( x, ˆ λ ) )). Then Theorem 4.1 says L ( ¯ G λ , A n − r × G rm / F q ( λ ) , T ) ( − n +1 = det (1 − ¯ α ˆ λ T )= (1 − π ( λ ) T ) · · · (1 − π υ S ( ¯ f ) ( λ ) T ) . For each λ , set A S ( λ ) := { π i ( λ ) } υ S ( ¯ f ) i =1 . Let L be a linear algebra operation. Define L ( LA S , G sm / F q , T ) := Y λ ∈| G sm / F q | Y τ ( λ ) ∈LA S ( λ ) (1 − τ ( λ ) T deg ( λ ) ) − . By a similar method to that of the previous section, we have: Theorem 4.2. For each linear algebra operation L , the L -function L ( LA S , G sm / F q , T ) is a rational function: L ( LA S , G sm / F q , T ) ( − s +1 = Q Ri =1 (1 − α i T ) Q Sj =1 (1 − β j T ) ∈ Q ( ζ p )( T ) . Further, if we let N ′ := υ S ( ¯ f ) then(a) the reciprocal zeros and poles α i and β j are algebraic integers, and for each j , β j = q n j α j for some positiveintegers n j .(b) If ˜ s < s then R = S , else if ˜ s = s then ≤ R − S ≤ s ! vol (Γ) L N ′ . (c) the total degree is bounded above by R + S ≤ L N ′ · ˜ s ! vol (Γ) · ˜ s +(1+ s ) n |L| (1 + 2 s ) s . d) If ¯ G ∈ F q [ x , . . . , x n , t , . . . , t s ] , then L ( LA S , A s / F q , T ) is a rational function, and writing L ( LA S , A s / F q , T ) = Q R ′ i =1 (1 − α i T ) Q S ′ j =1 (1 − β j T ) ∈ Q ( ζ p )( T ) , the zeros and poles satisfy ord q ( α i ) and ord q ( β j ) ≥ w (Γ) + w ( L B ( S ) ) , where w ( L B ( S ) ) is the minimumweight of the basis L B ( S ) (see (52)). Similar bounds to those found in Theorem 1.2 for degree and totaldegree may be given. Let λ ∈ ( F ∗ q ) s . In the case S = S , Adolphson and Sperber [5] observed that in the absolute case that the L -function of the exponential sum defined on A n by ¯ G ( x, λ ) was identified with the highest weight factor of the L -function of the exponential sum defined by the same ¯ G on G nm . This observation was generalized to moregeneral simplicial toric sums [6]. We give now a relative version of this highest weight factor result. Once morelet ¯ f ∈ F q [ x ± , . . . , x ± n ] with dim ∆ ∞ ( ¯ f ) = n . Let σ be the unique face of ∆ ∞ ( ¯ f ) containing the origin whichspans a linear subspace of smallest dimension; denote this dimension by dim ( σ ). If the origin is an interior pointof ∆ ∞ ( ¯ f ) then σ = ∆ ∞ ( ¯ f ). We say ∆ ∞ ( ¯ f ) is simplicial with respect to the origin if σ is contained in exactly n − dim ( σ ) faces of codimension n − ∞ ( ¯ f ). This is always satisfied when n = 2. It is also always truewhen the origin is an interior point of ∆ ∞ ( ¯ f ). Note that if ∆ ∞ ( ¯ f ) is simplicial with respect to the origin, thenthis holds as well for ¯ G .Using notation which is consistent with the previous section, we will write dim ( σ ) = n − r . Let the equationsof the hyperplanes { H i } ni = r +1 spanned by each of the codimension one faces { σ i } ni = r +1 of ∆ ∞ ( ¯ f ) containing theorigin be given by l i ( x , . . . , x n ) := n X j =1 a i,j x j = 0 r + 1 ≤ i ≤ n, where a i,j ∈ Z and for each r + 1 ≤ i ≤ n , gcd ( a i,r +1 , . . . , a i,n ) = 1. We assume the inequalities l i ( x , . . . , x n ) ≥ Cone ( ¯ f ). Let L be the greatest common divisor of all the ( n − r ) × ( n − r ) subdeterminants of the ( n − r ) × n integer matrix A := ( a i,j ) r +1 ≤ i ≤ n, ≤ j ≤ n . If p ∤ L then we may find n − r columns, say the last n − r columnsfor convenience of notation, so that the matrix A may be written in block form as ( A | A ) with A a square( n − r ) × ( n − r ) matrix with integer determinant which is relatively prime to p .Write S = { , . . . , n } , S = { , . . . , r } , and S = { r + 1 , . . . , n } . Let I be the r × r identity matrix and let˜ A := I A A , n × n matrix with entries ˜ a i,j , so ˜ a i,j = a i,j for i ≥ r + 1. For i ∈ S , write l i ( x , . . . , x n ) := n X j =1 ˜ a i,j x j and set D i,t := D i,t for i < r + 1 P nj =1 ˜ a ij D j,t for i ≥ r + 1 and ¯ D i,t = ¯ D i,t for i < r + 1 P nj =1 ¯ a ij ¯ D j,t for i ≥ r + 1 . (58)Define the complex Ω • ( C , ˜ ∇ ¯ G ) as follows. It has the same spaces as those of the complex Ω • ( C , ∇ ¯ G ) but withboundary map on Ω i defined by (58):˜ ∇ ¯ G ( η dx k x k ∧ · · · ∧ dx k i x k i ) := n X l =1 D l,t ( η ) dx l x l ! ∧ dx k x k ∧ · · · ∧ dx k i x k i . The reduction modulo π of this complex is the complex Ω • ( ¯ M , ˜ ∇ ¯ G ) with the same Ω i space as Ω • ( ¯ M , ∇ ¯ G ) butwith boundary map ˜ ∇ ¯ G ( η dx k x k ∧ · · · ∧ dx k i x k i ) := n X l =1 ¯ D l,t ( η ) dx l x l ! ∧ dx k x k ∧ · · · ∧ dx k i x k i . Theorem 4.3. If p ∤ L , then Ω • ( C , ˜ ∇ G ) and Ω • ( C , ∇ G ) are isomorphic as O -modules, and Ω • ( ¯ M , ˜ ∇ ¯ G ) and Ω • ( ¯ M , ∇ ¯ G ) are isomorphic as ¯ S -algebras. For A ⊂ S , we define C ( A )0 (and ¯ M ( A ) respectively) to be the ideal of elements in C (respectively in ¯ M ) withsupport in the monomials x µ = x µ · · · x µ n n such that l i ( µ ) ≥ i ∈ A . Then for A ⊂ S , l ∈ S , D l,t C ( A ∩S )0 ⊂ C (( A ∪{ l } ) ∩S )0 and ¯ D l,t ¯ M ( A ∩S ) ) ⊂ ¯ M (( A ∪{ l } ) ∩S ) . (59)We proceed in a manner entirely analogous to Section 4.1.In order to simplify the notation we will denote by S ˜Ω • ( C ) the subcomplex of Ω • ( C , ˜ ∇ ¯ G ) defined by S ˜Ω i := M A = { ≤ j < ··· Assume ¯ f is nondegenerate with respect to ∆ ∞ ( ¯ f ) and that ∆ ∞ ( ¯ f ) is simplicial with respect to theorigin. Assume p ∤ L . Then H i ( S ˜Ω • ( C )) and H i ( S ˜Ω • ( ¯ M )) are acyclic except in top dimension n . Furthermore, H n ( S ˜Ω • ( C )) is a free O -module of rank υ S ( ¯ f ) = X A ⊂S ( − | A | ( n − | A | )! vol A (∆ ∞ ( ¯ f A ))44 here here we let H A be the intersection of the hyperplanes { H i = 0 } for i ∈ A and vol A (∆ ∞ ( ¯ f A )) is the volumewith respect to Haar measure in H A normalized so that the fundamental domain for the lattice Z n ∩ H A in H A has measure 1.Let ¯ B ( S ) = S i ∈ D Z ≥ ¯ B ( S ,i ) where ¯ B ( S ,i ) is a subset of the monomials in ¯ R ( S ,i ) = ¯ R ( S ) ∩ ¯ R ( i ) such that ¯ V ( S ,i ) the F q -space spanned by ¯ B ( S ,i ) satisfies ¯ R ( S ,i ) = ¯ V ( S ,i ) M n X j =1 l j ( x ∂ ¯ f∂x , . . . , x n ∂ ¯ f∂x n ) ¯ R ( S −{ j } ,i ) (here S − { j } = S if i 6∈ S ). Then C ( S )0 ( 1 p − X v ∈ ¯ B ( S O π w Γ ( v ) v ⊕ n X j =1 D j,t C ( S −{ j } )0 ( 1 p − . We proceed in a manner quite analogous to the case in the previous section. For each λ ∈ ( F ∗ q ) s , let L ( ¯ G λ , G rm , T ) ( − n +1 = N Y i =1 (1 − π i ( λ ) T )with N := n ! vol ∆ ∞ ( ¯ f ). Set A ( λ ) = { π i ( λ ) } Ni =1 and let W n ( λ ) be the subset of A ( λ ) consisting of reciprocalzeros of highest archimedean weight: W n ( λ ) := { π ( λ ) ∈ A ( λ ) | | π ( λ ) | = q deg ( λ ) n/ } . The set W ( λ ) now plays precisely the same role as A S ( λ ) in the previous section. For a linear algebraic operation L , set L ( LW n , G sm / F q , T ) := Y λ ∈| G sm / F q | Y τ ( λ ) ∈LW n ( λ ) (1 − τ ( λ ) T deg ( λ ) ) − . Since Galois action preserves weight, this is a rational function over Q ( ζ p ) with properties as follows. Theorem 4.5. For each linear algebraic operation L , the L -function L ( LW n , G sm / F q , T ) is a rational functionover Q ( ζ p ) with estimates for its degree, total degree, and for the p -divisibility of its reciprocal zeros and poles areprecisely the same as those in Theorem 4.2. Here υ S ( ¯ f ) is the same alternating sum of volumes as in Theorem4.2 and here w ( LB ( S ) ) is the minimum of the weights in the basis LB ( S ) . L -function Let ¯ G ( x, t ) := ¯ f ( x ) + ¯ P ( x, t ) ∈ F q [ x ± , . . . , x ± n , t , . . . , t s ]where ¯ f ( x ) is nondegenerate with respect to ∆ ∞ ( ¯ f ). Let ¯ G satisfy the hypotheses of the toric family in Section 3,that is, dim ∆ ∞ ( ¯ f ) = n , ¯ f is nondegenerate with respect to ∆ ∞ ( ¯ f ), and 0 ≤ w ( µ ) < x µ in Supp ( ¯ P ).45or each λ ∈ F ∗ sq the L -function L ( ¯ G λ , Θ , G nm / F q ( λ ) , T ) has a unique unit root, say π ( λ ). Define the k -thmoment unit root L -function by L unit ( k, ¯ G, A s / F q , T ) := Y λ ∈| A s / F q | (1 − π ( λ ) k T deg ( λ ) ) − . This is a meromorphic function by Wan’s theorem [29, Theorem 8.4] and so may be written as L unit ( k, ¯ G, A s / F q , T ) = Q ∞ i =1 (1 − α i T ) Q ∞ j =1 (1 − β j T ) with α i , β j → i, j → ∞ . 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