Dwork crystals II

This paper is a continuation of [3], which we will refer to as Part I. We start with recalling our notations and definitions. Let p be a prime and R a p-adically complete characteristic zero domain such that ∩spR = {0}. Let f ∈ R[x 1 , . . . , x n ] be a Laurent polynomial and ∆ ⊂ R be its Newton polytope. A subset μ ⊂ ∆ is said to be open if its complement ∆\μ is a union of faces of any dimensions. For such a subset we consider the R-module of rational functions


Introduction
This paper is a continuation of [3], which we will refer to as Part I. We start with recalling our notations and definitions. Let p be a prime and R a p-adically complete characteristic zero domain such that 1 , . . . , x ±1 n ] be a Laurent polynomial and ∆ ⊂ R n be its Newton polytope. A subset µ ⊂ ∆ is said to be open if its complement ∆ \ µ is a union of faces of any dimensions. For such a subset we consider the R-module of rational functions When µ = ∆ we tend to omit it from the notation, e.g. Ω f (∆) is simply Ω f . The submodule of derivatives dΩ f ⊂ Ω f is defined as the R-span of all x i ∂ ∂x i ω with ω ∈ Ω f and 1 ≤ i ≤ n. In Part I we constructed, for every Frobenius lift σ on R, an R-linear Cartier operator on the p-adic completions This operator commutes with the derivations of R and satisfies C p • x i ∂ ∂x i = p x i ∂ ∂x i • C p for 1 ≤ i ≤ n. It is then immediate that the Cartier operator preserves dΩ f . We consider submodules U f (µ) = {ω ∈ Ω f (µ) | C s p (ω) ≡ 0 (mod p s Ω f σ s (µ)) for all s ≥ 1}. It follows from the above mentioned commutation relations that dΩ f ∩ Ω f (µ) ⊂ U f (µ). Denote by µ Z = µ ∩ Z n the set of integral points in µ. The main result of Part I states that if the Hasse-Witt matrix β p (µ) = coeffcient of x pv−u in f (x) p−1 u,v∈µ Z is invertible then the quotient is a free R-module of rank h = #µ Z where the images of can be taken as a basis. In this case, for every Frobenius lift σ and every derivation δ on R we define matrices Λ σ , N δ ∈ R h×h by the conditions One has Λ σ ≡ β p (µ) (mod p) and hence C p : Q f (µ) → Q f σ (µ) is invertible. In this paper we shall give explicit formulas for the matrices Λ σ , N δ in a number of situations. One p-adic approximation was already given in Part I: where β m (µ) ∈ R h×h is given by the same formula as the above Hasse-Witt matrix with p replaced by a positive integer m.
Let us say that a formal series for every s ≥ 1. In [4] Dwork proved this congruence for a class of hypergeometric series. His result was generalized in [5] for the generating series of sequences where g(x) is a multivariable Laurent polynomial such that its Newton polytope ∆ contains 0 as its only internal integral point. In Sections 2, 3 and 4 we shall apply our methods to give an alternative proof of the main result of [5]. Namely, with f (x) = 1 − tg(x) and µ = ∆ • the module Q f (µ) has rank 1 and we will see that Λ σ = q(t)/q(t p ). Dwork's congruence then follows from a p-adic approximation similar to (1), where In Section 4 we explore the relation between truncations and periods modulo m used in Part I; this relation is the key fact in our proof of Dwork's congruences. The main result of this paper is Theorem 12. It generalizes Dwork's congruences to the A-hypergeometric setting.
At the end of this introduction we would like to recall a detail from Part I which will be also useful for us here. When there is a vertex b ∈ ∆ such that the coefficient of f at b is a unit in R, one can give the following description of our Cartier operator. By expanding rational functions into formal power series supported in the cone The Cartier operation on formal expansions is simply given by Prop 8].

Periods
In part I we introduced the Cartier operator as operator on infinite Laurent series. However, the image of a rational function under the Cartier operator is again rational. Consider the rational function ω = g(x) f (x) k ∈ Ω f . We assert that the image of ω under C p is given by 1 p n y:y p =x where the summation is over all y = (ζ r 1 p x 1/p 1 , . . . , ζ rn p x 1/p n ) with 0 ≤ r 1 , . . . , r n < p, with ζ p a primitive p-th root of unity. This is again a rational function, but with denominator y:y p =x f (y) k . Choose a vertex b of the Newton polytope ∆ of f and expand in a Laurent series with respect to x b . The result is a Laurent series with support in the cone C(∆ − b). Suppose it reads k a k x k . Then application of C p yields The summation over the integers r 1 , . . . , r n yields something non-zero if and only if p divides k i for i = 1, . . . , n. The summation value then equals p n . Replacing k by pk then yields which is precisely the Cartier operator defined in Part I. There are also other ways to produce Laurent series expansions of ω. This happens in the case when R has another non-archimedean valuation, let us call it the t-adic valuation, and one coefficient of f that dominates all the others t-adically. So let us write f = w∈∆ Z v w x w and suppose that there exists v such that v v is a unit in R and |v v | t > |v w | t for all w = v. We can then expand ω in a t-adically converging Laurent series via The series expansion is t-adically convergent, but when v is not a vertex of ∆ we may end up with a Laurent series in x whose support is not a cone. It could possibly be all of Z n . The coefficients are then in the completion of R with respect to |.| t . We denote this completion by S and assume that v v ∈ S × . Suppose we get Assuming that for v 1 , v 2 ∈ R inequality |v 1 | t > |v 2 | t implies |σ(v 1 )| t > |σ(v 2 )| t , one can do analogous expansion in Ω f σ . Then, the same argument as above yields Then define the period map p v : Ω f → Sgiven by p v (ω) = c 0 , the constant term in the Laurent series expansion of ω with respect to v.
For a differential ring S with a homomorphism R → S which extends the derivations of R, a period map is an R-linear map p : Ω f → S which vanishes on dΩ f and commutes with derivations of R. Values of a period map on elements of Ω f are called periods. All period maps considered in this paper satisfy an extra condition of vanishing on the submodule of formal derivatives U f = Ω f ∩ dΩ formal . It follows almost from the definition that p v vanishes on dΩ f . It is slightly less trivial to see that p v vanishes on the formal derivatives. Proposition 2. Let notation be as above. Then, for all η ∈ U f we have p v (η) = 0.
Proof. First of all notice that the constant term of η equals the constant term of C s p (η) for all s ≥ 0. Since η ∈ U f we also know that the C s p (η) ≡ 0(mod p s ). In particular the constant term of η is divisible by p s for all s ≥ 0, hence equals 0. We conclude that p v (η) = 0. Theorem 3. Let µ ⊆ ∆ be an open set and h = #µ Z . Consider the column vector Assume that R is p-adically complete and the Hasse-Witt matrix β p (µ) is invertible in R. For any Frobenius lift σ and any derivation δ of R, we have and Proof. Consider the equality Expand all terms in a Laurent series with respect to the vertex v and determine the constant coefficient. Using the fact that the constant term of elements in U f vanish (Proposition 2) we get the first statement. In a similar vein, starting with we get the second statement again by taking the constant term of the Laurent series expansions with respect to v.

Example
Let g(x) be a Laurent polynomial in x 1 , . . . , x n with coefficients in Z p . Suppose that 0 is the only lattice point in the interior of the Newton polytope ∆ of g. We introduce another variable t and define f (x) = 1 −tg(x). We apply Theorem 3 to f (x) with µ = ∆ • and u = v = 0. In this case β m has only one entry, the constant coefficient of reads k≥0 b k t k with b k equal to the constant term of g(x) k . Take the Frobenius lift given by t → t p . Then we obtain as a consequence of Theorem 3, One easily checks that These can be interpreted as truncated version of the power series q(t). In [5] it is shown that This is a generalization of the famous congruence in Dwork's 'p-adic cycles' [4, (6.29)]. The latter can be obtained using g(x) = x + 1/x + y + 1/y. Note that Theorem 5 with γ m replaced by β m is simply Corollary 4. We shall prove Theorem 5 in the next section. It will follow from our proof that in fact q(t) q(t p ) ≡ γm(t) γ m/p (t p ) (mod p ordp(m) ) with any m ≥ 1, and a similar congruence holds for the derivatives: It is a curious fact that when g(x) has coefficients in Z then the series q is a p-adic analytic element for each p.

Truncations
In this section we consider periods mod m which, in a number of relevant cases, turn out to be truncations of the Laurent series solutions of a system of linear differential equations. But first we turn to general f (x) with coefficients in a p-adic ring R. By a period map mod m we mean an R-linear map ρ : Ω f → R such that ρ(dΩ f ) ⊂ mR and ρ•δ ≡ δ •ρ(mod mR) for every derivation δ on R. All period maps mod m considered in this paper will satisfy the condition ρ(U f ) ⊂ mR of "vanishing" on formal derivatives. Choose a vertex b ∈ ∆ and consider Laurent series expansions with respect to b. We assume its coefficient f b in f to be a unit in R. For an integer m ≥ 1 and a Laurent polynomial is a period map mod m. It is clear that on formal derivatives we also have ρ m,g (U f ) ⊂ mR. These properties follow easily if one observes that, modulo m, mth powers behave like constants under derivations (see Part I, Lemma 15). In Part I we already used two particular instances of these period maps: τ mv = ρ m,x −v f (x) for v ∈ ∆ Z and α mk = ρ m,x −k for k ∈ C(∆ − b) Z . We now describe their behaviour under the Cartier operator and relevant congruences in this more general context: Proposition 6. For a Laurent polynomial g = g w x w denote g σ = g σ w x w . For any m ≥ 1 divisible by p we have ρ m,g ≡ ρ m/p,g σ • C p (mod p ord p (m) ).
Proof. Similar to the proof of Proposition 16 in Part I.
Theorem 7. Let µ ⊆ ∆ be an open set and h = #µ Z . For m ≥ 1 consider column vectors ρ m ∈ R h with components ρ m,g (ω u ) for u ∈ µ Z . If R is p-adically complete and the Hasse-Witt matrix β p (µ) is invertible, then for any Frobenius lift σ and any derivation δ of R we have for all m ≥ 1.
Proof. Similar to the proof of Theorem 17 in Part I.
Let us choose a tuple of elements φ v ∈ R for v ∈ ∆ Z and consider matrices of periods mod m given by Observe that the entries of γ m do not depend on the choice of b since they are constant terms of Laurent polynomials that are independent of b. For a subset µ ⊂ ∆ we denote by γ m (µ) the submatrix given by (γ m ) u,v∈µ Z . We can rewrite these matrices via β-matrices as from which the following congruence follows trivially: Lemma 8. We have β p (µ) ≡ γ p (µ)(mod p). In particular β p (µ) is invertible if and only if this holds for γ p (µ).
Corollary 9. Let γ m (µ) be as above and suppose γ p (µ) is invertible. Then for any Frobenius lift σ and any derivation δ of R we have for all m ≥ 1.
As it follows from the first congruence in this corollary, we have Hence all γ p s (µ) are invertible and we obtain p-adic limit formulas Proof of Theorem 5. We apply Corollary 9 in the case f (x) = 1 − tg(x), φ = 1 and µ = ∆ • . Then γ m (µ) is the polynomial m−1 k=0 b k t k . It follows from Corollary 9 with σ(t) = t p that γ p s (t) ≡ Λγ p s−1 (t p )(mod p s ) for all s ≥ 1. Theorem 5 then follows from Corollary 4 which says that Λ = q(t)/q(t p ).

A-hypergeometric periods
We continue the calculation of periods following the idea in Section 2. Let f (x) = N i=1 v i x a i , where the v i are independent variables. This is the A-hypergeometric setting. Let ∆ ⊂ R n be the Newton polytope of f (x), which is now the convex hull of the set {a 1 , . . . , a N } ⊂ Z n . Pick some integer exponent vector u ∈ k∆, expand x u f (x) −k with respect to a i ∈ ∆ Z and take the constant term. We get Before we proceed we like to make a remark which considerably simplifies our calculation. Denote byã r ∈ Z n+1 the exponent vector a r preceded by an extra component 1. We call the set A = { a 1 , . . . , a N } ⊂ Z n+1 saturated when When A is saturated, the following Proposition can be applied to any exponent vector u: Proposition 10. For an integral point u ∈ k∆ we denoteũ = (k, u). Assume that there exist α 1 , . . . , α N ∈ Z ≥0 such that N r=1 α rãr =ũ. Then p a i (x u f (x) −k ) is equal to the application of the differential operator (−1) k−1 (k−1)! N r=1 ∂ αr r where ∂ r = ∂ ∂vr to the universal series The proof is straightforward with induction on k.
We proceed with the calculation of p a i (log f ) and get where the sum is over all non-negative ℓ 1 , . . . ∨ . . . ℓ N , not all zero, such that r =i ℓ r (a r − a i ) = 0. Here the ∨ in the summation range and the sum itself means that ℓ i is to be omitted. Introduce ℓ i = − r =i ℓ r . Recall our notationã r = (1, a r ). Then the definition of ℓ i sees to it that the support of the resulting Laurent series (aside from the constant log v i ) is contained in the set In order to have a more compact notation, let us rewrite the multinomial coefficient as where Γ * (n) with n ∈ Z is defined as (n − 1)! if n ≥ 1 and (−1) n /|n|! if n ≤ 0. Notice that the modified Γ * satisfies Γ * (n + 1) = nΓ * (n) for all integers n = 0. One also checks that Γ * (n)Γ * (1 − n) = sign(n)(−1) n−1 for all integers n. Here sign(n) = −1 if n ≤ 0 and 1 if n ≥ 1. The period now takes the shape Although we do not need this in the rest of this paper, we like to notice that this period is a Laurent series solution of the A-hypergeometric system of equations with A-matrix the matrix with columnsã 1 , . . . ,ã N and parameter vector −ũ. When we vary the different periods over i we see that the supports of the Laurent series also vary. Fortunately it turns out that their union also lies in a regular cone. The following result, as well as its proof, is taken from [1, Prop 2.9]. We use a different formulation however.
so we see thatã i is a (real) positive linear combination of some otherã k . Define the set C = {ã k |there exists j such that l (j) k = 0}. So C is the set ofã k that are non-trivially involved in some relation ℓ (j) . Suppose C is not empty. Letã k be a vertex of the convex hull of C. Suppose that l (k) k < 0. Thenã k , being a positive linear combination of otherã j ∈ C cannot be a vertex of the convex hull of C. So l (k) k ≥ 0 and fortiori, l (j) k ≥ 0 for all j. Their sum should be zero, contradicting the fact that l (j) k = 0 for some values of j. Hence we conclude that C is empty. In particular ℓ (j) = 0 for all j.
Due to Lemma 11 the set of formal power series supported in L • = L • (R) ∩ Z N is a ring. Let us denote this ring by We will also consider the bigger ring . Elements of S are power series supported in a finite number of integral translations of the cone L • . It follows from Proposition 10 and formula (10) that Note that when A is saturated, this argument can be applied with any k ≥ 1 and u ∈ k∆. With a bit more effort one can also show that p a i (x u f (x) −k ) ∈ S for any integral u ∈ k∆ without the assumption. In what follows we shall not assume that A is a saturated set. We shall be interested in the N × N matrix Ψ with entries This formula follows from (10) with u = a j and k = 1. It will be convenient to work with the renormalized seriesΨ ji := v j Ψ ji ∈ R. Let us now consider their truncated versions. Define for any m ≥ 1 the N × N-matrix ψ m with entries A straightforward calculation shows that this is equal to the series development (9) with k = 1, u = a j summed over m = 0, 1, 2, . . . , M − 1. Further calculation along the same lines as earlier shows that we get ℓ kãk = 0, ℓ k ≥ 0 for all k = i and ℓ i > −m}.
Comparing (12) and (11) one sees that (ψ m ) ji := v j (ψ m ) ji ∈ R is the truncation of the elementΨ ji = v j Ψ ji ∈ R. Let us consider the function | · | : L • → Z ≥0 given by ℓ k for ℓ ∈ L • and define truncations of elements of R by for all m ≥ 0. With this notation, the above computation shows thatΨ(m) =ψ m . Note that the constant termΨ(0) is the identity matrix, and henceΨ and all its truncations ψ m are invertible over R.
Theorem 12. Let µ ⊆ ∆ be an open set and denote h = #µ Z . Assume that h ≥ 1 and #{j : a j ∈ µ} = h. Consider the h × h submatrices with entries in R given bỹ whereΨ ji = v j Ψ ji are renormalized series (11). Letψ m =Ψ(m) for m ≥ 1 be the respective truncations. For the Frobenius lift σ : R → R that sends v j to v p j for each 1 ≤ j ≤ N and any of the derivations δ = v i ∂ ∂v i : R → R one has congruences for all m ≥ 1.
Let V be the h × h diagonal matrix with the entries v j for a j ∈ µ. Note that substituting Ψ = V Ψ andψ m = V ψ m into (13) and (14) shows that these congruences are equivalent to (mod p ordp(m) ). Matrices in the latter congruences have entries in the bigger ring S. We preferred to state our theorem for the normalized matrices because truncations are more naturally defined on elements of R rather than S.
Proof. Consider the matrices of periods mod m given by (8) with φ a i = v i : Their entries are in Z[v 1 , . . . , v N ] and we have γ m = V −1ψ m V m . It particular, the co- Since det(γ p ) is not divisible by p, this ring satisfies our assumption ∩ s≥1 p s R = {0} and hence one can apply Corollary 9. It follows that there are matrices Λ σ , N δ ∈ R h×h such that Observe that all matrices γ m are invertible over S because One of the consequences of this fact is that R is a subring of the p-adic completion . Working in the big ring S we can invert matrices in (16) and conclude that γ m · σ(γ m/p ) −1 ≡ Λ σ and δ(γ m ) · γ −1 m ≡ N δ (mod p ordp(m) ).
Substituting γ m = V −1ψ m V m in the left-hand sides yields ). One particular consequence of these congruences is that the matrices in their right-hand sides have entries in R. Secondly, they must coincide with the limits of the left-hand sides which, using the fact thatψ m is a truncation ofΨ, immediately implies that Substituting these values back into (17) proves our theorem.
The above proof is based on the ideas from Section 4. By Lemma 8 the Hasse-Witt matrix β p (µ) is congruent modulo p to the matrix γ p given in (15). (In the special case µ = ∆ • this was observed in [1, Proposition 3.8].) Using this fact we can conclude from the above proof that under the assumptions of Theorem 12 the determinant of the Hasse-Witt matrix is a polynomial not divisible by p and there exist the respective matrices Λ σ , N δ ∈ R h×h , where R is the p-adic completion of the ring Z[v ±1 1 , . . . , v ±1 N , det(β p (µ)) −1 ]. These are the same ring R and the same matrices that were used in the proof. In particular, R is a subring of the p-adic completion S = S and we have A special consequence of this corollary is that the matrices V Λ σ V −p and V N δ V −1 + δ(V )V −1 have their entries in R. Furthermore, it turns out that N δ and, in a lesser way, Λ σ , are independent of the choice of p. Finally, we remark that in fact there are well defined period maps p a i : Ω f → S.
As we explained in Section 2, these period maps are invariant under the Cartier operator (we have p a i = p σ a i • C p where p σ a i denotes the respective period map Ω f σ → S ) and vanish on formal derivatives. Corollary 13 is then a direct consequence of Theorem 3. Let us also mention the main result of [2], Theorem 1.4. It states that in the Ahypergeometric setting with the assumption that ∆ has a 0 as its unique interior lattice point the series Φ(v)/Φ(v p ), where Φ(v) = Ψ 00 (v 0 , . . . , v N ) is the unique entry of our matrix Ψ for µ = ∆ • , is a p-adic analytic element with the set of poles determined by the Hasse invariant β p (∆ • ). Hence [2, Theorem 1.4] follows from Corollary 13.

Example
We continue the example from Part I, Section 7 with We determine the entries of the matrixΨ. The vectorsã k are given by the columns of  The supports L i lie in the null space of this matrix which can be written as (r + 2s, s, s, r, −2r − 4s), r, s ∈ Z.
x r y s = 1 √ 1 − 4x F 1/4, 3/4, 1 64y The other components v j Ψ j,5 are not so easy to express in terms of one-variable hypergeometric functions, if possible at all.