Dwork crystals I

TL;DR

This paper provides an elementary explanation of Dwork's approach to explicit p-adic limit formulas for zeta functions associated with toric hypersurfaces, enhancing understanding of their p-adic properties.

Contribution

It offers a simplified, accessible presentation of Dwork's method for computing zeta functions of toric hypersurfaces using p-adic limits.

Findings

Explicit p-adic limit formulas for zeta functions derived

Enhanced understanding of Dwork's method for toric hypersurfaces

Elementary elaboration makes the theory more accessible

Abstract

We present an elementary elaboration of Dwork's idea of explicit $p$-adic limit formulas for zeta functions of toric hypersurfaces.

Equations

$$F(z)=\frac{1}{\pi }{\int }_1^{\infty }\frac{dx}{\sqrt{x(x-1)(x-z)}}=\sum _{k\ge 0}{\left(\frac{(1/2{)}_k}{k!}\right)}^2{z}^k$$

$$F_{p^s}(z)=\sum _{k=0}^{p^s-1}{\left(\frac{(1/2{)}_k}{k!}\right)}^2{z}^k.$$

$$F_{p^{s+1}}(z_0)/F_{p^s}(z_0)\equiv F_{p^s}(z_0)/F_{p^{s-1}}(z_0)(\mathrm{mod}{p}^s).$$

$$G_{p^{s}}(z)=\mbox{coefficient of $(xy)^{p^{s}-1}$ of }(y^{2}-x(x-1)(x-z))^{p^{s}-1}.$$

$$G_{p^{s+1}}(z_0)/G_{p^s}(z_0)\equiv G_{p^s}(z_0)/G_{p^{s-1}}(z_0)(\mathrm{mod}{p}^s)$$

$$(\beta_{m})_{{\bf u},{\bf v}\in J}:=\mbox{coefficient of ${\bf x}^{m{\bf v}-{\bf u}}$ of }f({\bf x})^{m-1}.$$

$${\beta }_{p^{s+1}}\sigma ({\beta }_{p^s}{)}^{-1}\equiv {\beta }_{p^s}\sigma ({\beta }_{p^{s-1}}{)}^{-1}(\mathrm{mod}{p}^s)$$

$$\delta ({\beta }_{p^{s+1}}){\beta }_{p^{s+1}}^{-1}\equiv \delta ({\beta }_{p^s}){\beta }_{p^s}^{-1}(\mathrm{mod}{p}^s)$$

$${\Lambda }_{\sigma }=\underset{s\to \infty }{\lim }{\beta }_{p^s}\sigma ({\beta }_{p^{s-1}}{)}^{-1}\text{mbox}and{N}_{\delta }=\underset{s\to \infty }{\lim }\delta ({\beta }_{p^s}){\beta }_{p^s}^{-1}.$$

$$(u_0-1)!\frac{{\mathbf{x}}^{\mathbf{u}}}{f(\mathbf{x}{)}^{u_0}},$$

$${\mathcal{W}}_f:={\Omega }_f^n/d({\Omega }_f^{n-1}).$$

$$C(\Delta )=\{\lambda (1,u_1,\dots ,u_n)|(u_1,\dots ,u_n)\in \Delta ,\lambda \ge 0\},$$

$${\omega }_{\mathbf{u}}:=(u_0-1)!\frac{{\mathbf{x}}^{\mathbf{u}}}{f(\mathbf{x}{)}^{u_0}}$$

$${\mathcal{W}}_f:={\Omega }_f/d{\Omega }_f,$$

$$k!\frac{g(\mathbf{x})}{f^{k+1}}\frac{d\mathbf{x}}{\mathbf{x}}$$

$$\frac{g(\mathbf{x})}{f(\mathbf{x}{)}^m}=\frac{g(\mathbf{x}){\mathbf{x}}^{-m\mathbf{b}}}{f_{\mathbf{b}}^m(1+\dots {)}^m}=\frac{g(\mathbf{x}){\mathbf{x}}^{-m\mathbf{b}}}{f_{\mathbf{b}}^m}\sum _{k\ge 0}{h}_k(\mathbf{x}),$$

$${\Omega }_{\mathrm{formal}}=\{\sum _{\mathbf{k}\in C(\Delta -\mathbf{b})}{a}_{\mathbf{k}}{\mathbf{x}}^{\mathbf{k}}|a_{\mathbf{k}}\in R\}.$$

$$a_{{\bf k}}\equiv 0({\rm mod}\ \gcd(k_{1},\ldots,k_{n}))\quad\mbox{for all ${\bf k}$}.$$

$$x_i\frac{\partial }{\partial x_i}\left(a{\mathbf{x}}^{\mathbf{k}}\right)=a{k}_i{\mathbf{x}}^{\mathbf{k}}\equiv 0(\mathrm{mod}{k}_i).$$

$$\sum _i{m}_i{x}_i\frac{\partial {\mathbf{x}}^{\mathbf{k}}}{\partial x_i}=\gcd (k_1,\dots ,k_n){\mathbf{x}}^{\mathbf{k}}.$$

$${\mathcal{C}}_p\left(\sum _{\mathbf{k}}{a}_{\mathbf{k}}{\mathbf{x}}^{\mathbf{k}}\right):=\sum _{\mathbf{k}}{a}_{p\mathbf{k}}{\mathbf{x}}^{\mathbf{k}}.$$

$${\mathrm{ord}}_p(r)=\sup \{s\in {\mathbb{Z}}_{\ge 0}:r\in p^{s}R\}$$

$${\mathcal{C}}_p:d{\Omega }_{\mathrm{formal}}\to pd{\Omega }_{\mathrm{formal}}.$$

$$\frac{{\mathbf{x}}^{\mathbf{u}}}{f(\mathbf{x}{)}^{u_0}}=\frac{{\mathbf{x}}^{\mathbf{u}}f(\mathbf{x}{)}^{p\lceil u_0/p\rceil -u_0}}{(f^{\sigma }({\mathbf{x}}^p)-pG(\mathbf{x}){)}^{\lceil u_0/p\rceil }}=\sum _{r\ge 0}{p}^r\left(\genfrac{}{}{0ex}{}{\lceil u_0/p\rceil +r-1}{r}\right)\frac{G(\mathbf{x}{)}^r}{f^{\sigma }({\mathbf{x}}^p{)}^{r+\lceil u_0/p\rceil }}{\mathbf{x}}^{\mathbf{u}}f(\mathbf{x}{)}^{p\lceil u_0/p\rceil -u_0}.$$

$${\mathcal{C}}_p({\omega }_{\mathbf{u}})=\sum _{r\ge 0}\frac{p^r}{r!}\frac{(u_0-1)!}{(\lceil u_0/p\rceil -1)!}(\lceil u_0/p\rceil +r-1)!\frac{Q_r(\mathbf{x})}{f^{\sigma }(\mathbf{x}{)}^{r+\lceil u_0/p\rceil }},$$

$${\mathcal{C}}_p({\omega }_{\mathbf{u}})=\sum _{\mathbf{v}\in C(\Delta {)}_{\mathbb{Z}}^{+}}{F}_{\mathbf{u},\mathbf{v}}{\omega }_{\mathbf{v}}^{\sigma },$$

$$F_{\mathbf{u},\mathbf{v}}=\{\begin{array}{ll}\frac{p^r}{r!}\frac{(u_0-1)!}{(\lceil u_0/p\rceil -1)!}\times \text{ coefficient of }{\mathbf{x}}^{\mathbf{v}}\text{ in }{Q}_r(\mathbf{x}),& r:=v_0-\lceil u_0/p\rceil \ge 0,\\ 0,& v_0<\lceil u_0/p\rceil .\end{array}$$

$${\mathrm{ord}}_p(F_{\mathbf{u},\mathbf{v}})\ge r-{\mathrm{ord}}_p(r!)+{\mathrm{ord}}_p\left(\frac{(u_0-1)!}{(\lceil u_0/p\rceil -1)!}\right).$$

$${\mathrm{ord}}_p(F_{\mathbf{u},\mathbf{v}})\ge r+\lceil u_0/p\rceil -1-\frac{r}{p-1}=v_0-1-\frac{r}{p-1}\ge \frac{p-2}{p-1}(v_0-1).$$

$${\mathcal{C}}_p:{\hat{\Omega }}_f\to {\hat{\Omega }}_{f^{\sigma }}.$$

Full text

Dwork crystals I

Frits Beukers, Masha Vlasenko

Utrecht University

f.beukers@uu.nl

Institute of Mathematics of the Polish Academy of Sciences

m.vlasenko@impan.pl

Work of Frits Beukers was supported by the Netherlands Organisation for Scientific Research (NWO), grant TOP1EW.15.313. Work of Masha Vlasenko was supported by the National Science Centre of Poland (NCN), grant UMO-2016/21/B/ST1/03084.

1. Introduction

In his study of zeta-functions of families of algebraic varieties Dwork discovered a number of remarkable congruences for truncated solutions of Picard–Fuchs equations. For example, let

[TABLE]

be the period function associated to the Legendre family of elliptic curves $y^{2}=x(x-1)(x-z)$. Here $(1/2)_{k}$ denotes the Pochhammer symbol $\Gamma(k+1/2)/\Gamma(1/2)$. Let $p$ be an odd prime and $s$ a positive integer. Let $F_{p^{s}}$ be the truncation of $F$ given by

[TABLE]

Let $z_{0}$ be a $p$-adic integer and suppose $F_{p}(z_{0})$ is a $p$-adic unit. Then $F_{p^{s}}(z_{0})$ is a $p$-adic unit for all $s\geq 1$ and we have

[TABLE]

The $p$-adic unit $\lambda(z_{0})=(-1)^{\frac{p-1}{2}}\underset{s\to\infty}{\lim}F_{p^{s}}(z_{0})/F_{p^{s-1}}(z_{0})$ is a root of the zeta function of the elliptic curve corresponding to $z_{0}({\rm mod}\ p)$. (Though it looks sligtly different, this fact is a version of [3, (6.29)].)

In a series of papers culminating in [8, Theorem 6.2] Katz developed a general theory of such congruences and their underlying mechanism. However, his congruences involve formal expansion coefficients of differential forms instead of truncated power series solutions of a differential equation. In this paper we consider a third alternative, namely coefficients of certain powers of the polynomial defining a variety. For example, in the case of the Legendre elliptic curve they are given by

[TABLE]

Although different from $F_{p^{s}}(z)$, they both satisfy the hypergeometric differential equation modulo $p^{s}$. The congruences read

[TABLE]

and the quotients converge to the $p$-adic unit root $\lambda(z_{0})$. In this paper we shall deal with a generalized version of the congruences of the latter type. A number of ideas in this paper are already present in [8], but in a very different language. There will also be no smoothness assumptions on the underlying variety. We plan to come back to the case of truncated power series solutions in a later paper.

Let $R$ be a ring of characteristic zero and $p$ a prime number. Suppose that we have a $p$th power Frobenius lift on $R$ which is a ring endomorphism $\sigma:R\to R$ with the property that $\sigma(r)\equiv r^{p}({\rm mod}\ p)$ for all $r\in R$. For example, when $R={\mathbb{Z}}$ is the ring of integers we can take $\sigma(r)=r$ for all $r$. When $R={\mathbb{Z}}[t]$ is a polynomial ring we can take $\sigma(g(t))=g(t^{p})$.

Let $f({\bf x})=\sum_{i=1}^{N}f_{i}{\bf x}^{{\bf a}_{i}}$ be a Laurent polynomial in $x_{1},\ldots,x_{n}$ with $f_{i}\in R$ for all $i$. Here we use the vector notation ${\bf x}^{{\bf e}}=x_{1}^{e_{1}}x_{2}^{e_{2}}\cdots x_{n}^{e_{n}}$. Let $\Delta\subset{\mathbb{R}}^{n}$ be the Newton polytope of $f({\bf x})$, which is the convex hull of its support $\{{\bf a}_{i}:f_{i}\neq 0\}$. Let $J$ be the set of interior lattice points in $\Delta$ and set $g=\#J$. We assume that $g>0$. For any integer $m\geq 1$ we define the $g\times g$-matrix $\beta_{m}$ with entries

[TABLE]

When $m=1$ we take for $\beta_{m}$ the identity matrix. We call $\beta_{p}$ the Hasse–Witt matrix of $f$. When $\beta_{p}$ is invertible modulo $p$ it turns out that $\beta_{p^{s}}$ is invertible modulo $p$ for every $s\geq 1$. Note that being invertible modulo $p$ implies being invertible modulo all powers of $p$.

In [11] it is shown that if the Hasse–Witt matrix is invertible modulo $p$, then

[TABLE]

for every $s\geq 1$. One may observe that this congruence as similar to the last part of Theorem 6.2 in Katz’s paper [8]. We believe that the merit of [11] is that the proof of the congruence is completely elementary.

Let $\delta$ be a derivation on $R$. Again in an elementary way, it is shown in [11] that if $\beta_{p}$ is invertible in $R$, then

[TABLE]

for every $s\geq 1$.

These congruences imply the existence, for each Frobenius lift $\sigma$ and each derivation $\delta$ on $R$, of $p$-adic limit matrices $\Lambda_{\sigma}$ and $N_{\delta}$ such that

[TABLE]

It is the goal of the present paper to give an interpretation of these matrices in terms of operations with regular rational functions on ${\mathbb{T}}^{n}\setminus Z_{f}$, the complement of the set of zeroes $Z_{f}=\{{\bf x}:f({\bf x})=0\}$ in the $n$-dimensional torus ${\mathbb{T}}^{n}$. At the same time we provide an alternative proof of the congruences.

To be slightly more precise, we consider the $R$-module $\Omega_{f}$ of rational functions generated over $R$ by

[TABLE]

where $u_{0}$ is a positive integer and ${\bf u}\in(u_{0}\Delta)\cap{\mathbb{Z}}^{n}$. Any derivation $\delta$ on $R$ can be extended naturally to $\Omega_{f}$ by setting $\delta(x_{i})=0$ for all $i$.

In this paper we construct the $R$-linear Cartier operator $\mathscr{C}_{p}:\widehat{\Omega}_{f}\to\widehat{\Omega}_{f^{\sigma}}$, where $\widehat{\Omega}_{f}=\underset{\leftarrow}{\lim}(\Omega_{f}/p^{s}\Omega_{f})$ is the $p$-adic completion of $\Omega_{f}$ and $f^{\sigma}({\bf x})=\sum_{i=1}^{N}f_{i}^{\sigma}\,{\bf x}^{{\bf a}_{i}}$ is simply $f$ with $\sigma$ applied to its coefficients. The Cartier operator commutes with any derivation $\delta$ of $R$.

The main results of this paper are Theorems 4.3 and 5.3. Applied to the open set $\mu=\Delta^{\circ}$ of interior points of $\Delta$, they describe a free rank $g$ subquotient $Q_{f}=Q_{f}(\Delta^{\circ})$ of $\Omega_{f}$ to which the Cartier operator descends and $\Lambda_{\sigma}$ is the (transposed) matrix that corresponds to the $R$-linear map $\mathscr{C}_{p}:Q_{f}\to Q_{f^{\sigma}}$. As a bonus of our considerations we also recover a version of Katz’s theorem [8, Theorem 6.2] as Theorem 5.7.

Finally in this introduction we point out the connection with the de Rham cohomology of the complement of $Z_{f}$. Define the modules $\Omega_{f}^{n}=\Omega_{f}{dx_{1}\over x_{1}}\wedge\cdots\wedge{dx_{n}\over x_{n}}$ and $\Omega_{f}^{n-1}=\oplus_{i=1}^{n}\Omega_{f}{dx_{1}\over x_{1}}\wedge\cdots\check{{dx_{i}\over x_{i}}}\cdots\wedge{dx_{n}\over x_{n}}$ of differential $n$- and $n-1$-forms respectively. The above mentioned $R$-module $Q_{f}$ is in fact a ($p$-adic) subquotient of

[TABLE]

We call the latter the Dwork module. It is known due to the work of Griffiths and Batyrev that, when $R$ is a field and $f$ satisfies certain regularity conditions (so called $\Delta$-regularity), then $\mathscr{W}_{f}$ is isomorphic to the middle de Rham cohomology $H^{n}_{dR}({\mathbb{T}}^{n}\setminus Z_{f})$ (see Corollary B.4 and [1, Theorem 7.13]). In particular, it is a vector space over $R$ of finite dimension. In this paper we will not assume regularity. We also will not assume that the Newton polytope $\Delta\subset{\mathbb{R}}^{n}$ is of maximal dimension.

2. Regular functions and formal expansion

Let $R$ be a characteristic zero ringdomain, $f\in R[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ be a Laurent polynomial and $\Delta\subset{\mathbb{R}}^{n}$ be its Newton polytope. By $C(\Delta)$ we denote the subset of ${\mathbb{R}}^{n+1}$ given by

[TABLE]

the positive cone spanned by the Newton polytope $\Delta$ placed in ${\mathbb{R}}^{n+1}$ in the hyperplane $u_{0}=1$.

The set of integral points $C(\Delta)\cap{\mathbb{Z}}^{n+1}$ is denoted by $C(\Delta)_{\mathbb{Z}}$. Let $C(\Delta)^{+}_{\mathbb{Z}}=C(\Delta)_{\mathbb{Z}}\setminus\{{\bf 0}\}$ be the set of non-zero integral points in the cone. For any $(u_{0},u_{1},\ldots,u_{n})={\bf u}\in C(\Delta)_{\mathbb{Z}}$ we denote ${\bf x}^{{\bf u}}=x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}$ (we simply drop the component $u_{0}$ here, as there is no respective variable $x_{0}$). Consider the $R$-module $\Omega_{f}$ of regular rational functions generated over $R$ by

[TABLE]

for all ${\bf u}\in C(\Delta)_{\mathbb{Z}}^{+}$. Note that $1$ is an $R$-linear combination of $\omega_{{\bf u}}$ with $u_{0}=1$, so the constant functions are also in $\Omega_{f}$.

We define the module $d\Omega_{f}$ as the $R$-span of all derivatives $x_{i}\frac{\partial}{\partial x_{i}}\omega$ with $\omega\in\Omega_{f}$ and $1\leq i\leq n$. Note that $d\Omega_{f}\subset\Omega_{f}$. The quotient $R$-module

[TABLE]

will be called the Dwork module.

Remark 2.1**.**

Having the extra factor $(u_{0}-1)!$ in the definition of $\omega_{{\bf u}}$ appears to be essential in many ways when working over rings $R$ (rather than fields). At the end of the introduction we mentioned that the Dwork module can be also written in terms of differential forms as $\mathscr{W}_{f}=\Omega^{n}_{f}/d(\Omega^{n-1}_{f})$. Factors $(u_{0}-1)!$ in $\omega_{{\bf u}}$ allow the so-called Griffiths–Dwork reduction when we work in $\mathscr{W}_{f}$. This is the procedure to reduce the pole order of a form by shifting it by exact forms. More concretely, suppose we have a form of the shape $k!\frac{g({\bf x})}{f^{k+1}}\frac{d{\bf x}}{{\bf x}}$ and there exist Laurent polynomials $g_{0}({\bf x}),g_{1}({\bf x}),\ldots,g_{n}({\bf x})$ with support in $k\Delta$ such that $g=g_{0}f+\sum_{i=1}^{n}g_{i}x_{i}\frac{\partial f}{\partial x_{i}}$. Then

[TABLE]

The final form is again in $\Omega_{f}^{n}$. Note that factorials appear in the Laplace transform in [1, §7].

Rational functions can be expanded as formal Laurent series. To that end we fix a vertex ${\bf b}$ of $\Delta$ and assume that the coefficient of $f$ at ${\bf x}^{\bf b}$ is a unit in $R$. Denote this coefficient by $f_{{\bf b}}$ and expand rational functions as

[TABLE]

where $h_{k}({\bf x})$ are Laurent polynomials supported in $k(\Delta-{\bf b})$ for every $k$. There are only finitely many summands contributing to each monomial in the cone $C(\Delta-{\bf b})\subseteq{\mathbb{R}}^{n}$. Observe that when $g({\bf x})$ is supported in $m\Delta$ the formal series in the right-hand side is itself supported in $C(\Delta-{\bf b})$. (Here we need a word of caution regarding our notation. In (1) the polytope $\Delta$ was placed in the hyperplane $u_{0}=1$ in ${\mathbb{R}}^{n+1}$, which will be our usual convention throughout the paper. Note that, with this convention, the difference $\Delta-{\bf b}$ is a polytope in the hyperplane $u_{0}=0$ and one can view the respective cone $C(\Delta-{\bf b})$ as a subset of this hyperplane $\{u_{0}=0\}\cong{\mathbb{R}}^{n}$.) Denote the ring of formal Laurent series with support in $C(\Delta-{\bf b})$ and coefficients in $R$ by

[TABLE]

It is indeed a ring because the cone has ${\bf 0}$ as a vertex. The above explained procedure of formal expansion defines an embedding of $\Omega_{f}$ into $\Omega_{\rm formal}$ as an $R$-submodule. Note that we do not include the choice of ${\bf b}$ in the notation $\Omega_{\rm formal}$.

Similarly to $d\Omega_{f}$, the $R$-module of formal derivatives $d\Omega_{\rm formal}$ is defined as the $R$-span of derivatives $x_{i}\frac{\partial}{\partial x_{i}}\omega$ with $1\leq i\leq n$ and $\omega\in\Omega_{\rm formal}$.

Lemma 2.2**.**

A series $\sum_{{\bf k}\in C(\Delta-{\bf b})}a_{{\bf k}}{\bf x}^{{\bf k}}$ is a formal derivative if and only if

[TABLE]

Proof.

Notice that for any monomial ${\bf x}^{{\bf k}}$, any $i$ and any $a\in R$ we have

[TABLE]

This shows the $\Rightarrow$ part. To see the reverse implication, write $\gcd(k_{1},\ldots,k_{n})=\sum_{i}m_{i}k_{i}$ for some $m_{i}\in{\mathbb{Z}}$ and note that

[TABLE]

∎

3. Cartier operator

Let us fix a prime number $p$. We define the Cartier operator $\mathscr{C}_{p}$ on $\Omega_{\rm formal}$ by

[TABLE]

Though acting on different spaces, this operation was already used in early papers of Dwork (see $\psi$ in [4, §2]) and Reich (see $\Psi$ in [9, §(b)]).

From now on we assume that $\cap_{s}p^{s}R=\{0\}$, in which case we have a well defined $p$-adic valuation

[TABLE]

on $R$ which extends the usual $p$-adic valuation on ${\mathbb{Z}}\subset R$. This valuation takes finite values on all non-zero elements of $R$ and satisfies the inequalities $\mathrm{ord}_{p}(r_{1}r_{2})\geq\mathrm{ord}_{p}(r_{1})+\mathrm{ord}_{p}(r_{2})$ and $\mathrm{ord}_{p}(r_{1}+r_{2})\geq\min(\mathrm{ord}_{p}(r_{1}),\mathrm{ord}_{p}(r_{2}))$. We also assume that $R$ is $p$-adically complete. In particular, $R$ is a ${\mathbb{Z}}_{p}$-algebra and Lemma 2.2 can be reformulated as

Lemma 3.1**.**

A series $h\in\Omega_{\rm formal}$ is a formal derivative if and only if $\mathscr{C}_{p}^{s}(h)\equiv 0({\rm mod}\ p^{s})$ for all integers $s\geq 1$.

One easily shows that $\mathscr{C}_{p}\circ\theta_{i}=p\,\theta_{i}\circ\mathscr{C}_{p}$ for any $\theta_{i}=x_{i}\frac{\partial}{\partial x_{i}}$. We thus observe that the Cartier operator preserves the submodule of formal derivatives and is divisible by $p$ on it, i.e.

[TABLE]

Applying this commutation identity $s$ times yields $\mathscr{C}_{p}^{s}\circ\theta_{i}=p^{s}\theta_{i}\circ\mathscr{C}_{p}^{s}$, which immediately gives one of the implications in the last lemma. Here is another straightforward property of $\mathscr{C}_{p}$:

Lemma 3.2**.**

Let $g,h\in\Omega_{\rm formal}$. Then $\mathscr{C}_{p}(g({\bf x}^{p})h)=g({\bf x})\mathscr{C}_{p}(h)$.

Since Cartier operators are usually defined mod $p$ in the literature, naming $\mathscr{C}_{p}$ a Cartier lift might be more appropriate. Nevertheless we prefer to call it the Cartier operator.

We now like to restrict the Cartier operator to $\Omega_{f}$. We will need the $p$-adic completion $\widehat{\Omega}_{f}:=\underset{\leftarrow}{\lim}\Omega_{f}/p^{s}\Omega_{f}$. Fix a Frobenius lift $\sigma$ on $R$: this is a ring endomophism $\sigma:R\to R$ such that $\sigma(r)\equiv r^{p}({\rm mod}\ p)$ for every $r\in R$. Our main observation is that

Proposition 3.3**.**

If $p>2$ then $\mathscr{C}_{p}(\Omega_{f})\subset\widehat{\Omega}_{f^{\sigma}}$.

Proof.

To see this, rewrite $1/f({\bf x})^{u_{0}}$ as $f({\bf x})^{p\lceil u_{0}/p\rceil-u_{0}}/f({\bf x})^{p\lceil u_{0}/p\rceil}$. Then note that $f({\bf x})^{p}=f^{\sigma}({\bf x}^{p})-pG({\bf x})$ for some Laurent polynomial $G$ with coefficients in $R$ and support in $p\Delta$. Then we use the $p$-adic expansion

[TABLE]

Multiply this with $(u_{0}-1)!$ and apply $\mathscr{C}_{p}$. Using Lemma 3.2 we find that

[TABLE]

where the $Q_{r}({\bf x})=\mathscr{C}_{p}(G({\bf x})^{r}{\bf x}^{{\bf u}}f({\bf x})^{p\lceil u_{0}/p\rceil-u_{0}})$ are Laurent polynomials in $x_{1},\ldots,x_{n}$ with support in $(\lceil u_{0}/p\rceil+r)\Delta$ and coefficients in $R$. The last formula can be rewritten as

[TABLE]

where

[TABLE]

To show that $\mathscr{C}_{p}(\omega_{{\bf u}})\in\widehat{\Omega}_{f^{\sigma}}$ it suffices to show that $\mathrm{ord}_{p}(F_{{\bf u},{\bf v}})\to\infty$ as $v_{0}\to\infty$. To that end we observe that

[TABLE]

It is straightforward to see that $\frac{(u_{0}-1)!}{(\lceil u_{0}/p\rceil-1)!}$ has order $\geq\lceil u_{0}/p\rceil-1$ and that $\mathrm{ord}_{p}(r!)<\frac{r}{p-1}$. This gives us

[TABLE]

The latter goes to $\infty$ with $v_{0}$ when $p>2$. ∎

From now on we assume that $p>2$. By Proposition 3.3 we have a well-defined $R$-linear map

[TABLE]

It is in fact given by the explicit formulas (4) and (5), which also show that the map (7) is independent of the choice of vertex ${\bf b}$ of $\Delta$ at which we are doing formal expansions.

We shall also be interested in regular functions supported in subsets of the cone $C(\Delta)$. For a subset $\mu\subseteq\Delta$ let us denote by

[TABLE]

the $R$-module generated by functions $\omega_{{\bf u}}$ with ${\bf u}\in C(\mu)_{\mathbb{Z}}^{+}$. Here $C(\mu)\subseteq C(\Delta)$ is the positive cone spanned by $\mu$ placed in ${\mathbb{R}}^{n+1}$ in the hyperplane $u_{0}=1$, and $C(\mu)_{\mathbb{Z}}=C(\mu)\cap{\mathbb{Z}}^{n+1}$ is the set of integral points in this cone and $C(\mu)_{\mathbb{Z}}^{+}=C(\mu)_{\mathbb{Z}}\setminus\{{\bf 0}\}$ is the set of non-zero integral points. Note that $\Omega_{f}(\Delta)=\Omega_{f}$. The respective $p$-adic completion is denoted $\widehat{\Omega}_{f}(\mu)=\underset{\leftarrow}{\lim}\;\Omega_{f}(\mu)/p^{s}\Omega_{f}(\mu)$.

Proposition 3.4**.**

Define a finite topology on $\Delta$, where the closed sets are unions of faces of any dimension. Let $\mu\subset\Delta$ be an open set. Then the Cartier operator $\mathscr{C}_{p}$ maps $\widehat{\Omega}_{f}(\mu)$ into $\widehat{\Omega}_{f^{\sigma}}(\mu)$ and derivations of $R$ map $\Omega_{f}(\mu)$ to itself.

Proof.

Since open sets are intersections of the complements of faces, it is enough to prove our statement for $\mu$ being such a complement. Without loss of generality we assume that $\mu^{c}$ is a face of $\Delta$. In this case $C(\mu)=\Sigma^{c}=C(\Delta)\setminus\Sigma$ where $\Sigma=C(\mu^{c})$ is the respective face of the cone $C(\Delta)$. The $R$-module $\Omega_{f}(\mu)$ is generated by functions $\omega_{\bf u}$ with ${\bf u}\in\Sigma^{c}$. To prove our proposition we recall that the Cartier operator (7) is given explicitly by formula (4) and one easily sees that ${\bf u}\in\Sigma^{c}$ and $F_{{\bf u},{\bf v}}\neq 0$ imply ${\bf v}\in\Sigma^{c}$.

Let $\delta$ be a derivation of $R$ and ${\bf u}\in C(\mu)_{\mathbb{Z}}$. Observe that in the formula

[TABLE]

the support of ${\bf x}^{{\bf u}}\delta(f)$ is in ${\bf u}+\Delta$, which again lies in $C(\mu)$ when $\mu$ is open in our sense. ∎

Definition 3.5**.**

Fix a non-empty subset $\mu\subseteq\Delta$ which is open in the topology from Proposition 3.4. Let $\mu_{\mathbb{Z}}=\mu\cap{\mathbb{Z}}^{n}$ be the set of integral points in $\mu$. We assume this set is non-empty and let $h=\#\mu_{\mathbb{Z}}$ be the number of such points. For any integer $m\geq 1$ we define the $h\times h$-matrix $\beta_{m}=\beta_{m}(\mu)$ with entries

[TABLE]

When $m=1$ we take for $\beta_{1}(\mu)$ the identity matrix. We call $\beta_{p}(\mu)$ the Hasse–Witt matrix of $f$ relative to $\mu$.

Proposition 3.6**.**

Suppose that $p>2$ and $\mu\subseteq\Delta$ is a non-empty subset which is open in the topology defined in Proposition 3.4. Then

[TABLE]

Moreover, for any ${\bf u}\in\mu_{\mathbb{Z}}$ we have

[TABLE]

Proof.

From the proof of Propositions 3.3 and 3.4, in particular equations (4) and (5), we know an expression for $\mathscr{C}_{p}(\omega_{{\bf u}})$ as a linear combination $\sum_{{\bf v}\in C(\mu)_{\mathbb{Z}}^{+}}F_{{\bf u},{\bf v}}$ for every ${\bf u}\in\mu_{\mathbb{Z}}$. Moreover, it follows from (6) that $F_{{\bf u},{\bf v}}\equiv 0({\rm mod}\ p)$ when $v_{0}>1$. Our first statement follows immediately. The observation that $(\beta_{p})_{{\bf u},{\bf v}}\equiv F_{{\bf u},{\bf v}}({\rm mod}\ p)$ whenever $v_{0}=1$ proves the second statement. ∎

4. The unit-root crystal

In this section we formulate the first main result of this paper. But first we need some preparations.

Definition 4.1**.**

For a non-empty subset $\mu\subseteq\Delta$ which is open in the topology from Proposition 3.4, define $U_{f}(\mu)=\widehat{\Omega}_{f}(\mu)\cap d\Omega_{\rm formal}$.

We call $U_{f}(\mu)$ the submodule of formal derivatives. Differential $n$-forms associated to elements of $U_{f}(\mu)$ were called forms that ’die on formal expansion’ by Nick Katz in [8, p.258]. It turns out that one can give a characterization of $U_{f}(\mu)$ which does not make any reference to $\Omega_{\rm formal}$:

Proposition 4.2**.**

With the notations as above we have

[TABLE]

Proof.

Let $\omega\in\widehat{\Omega}_{f}(\mu)$. Suppose that $\mathscr{C}_{p}^{s}(\omega)\equiv 0({\rm mod}\ p^{s}\widehat{\Omega}_{f^{\sigma^{s}}}(\mu))$ for all $s\geq 1$. Then it follows from Lemma 3.1 that $\omega\in d\Omega_{\rm formal}$.

Suppose conversely that $\omega\in\widehat{\Omega}_{f}(\mu)\cap d\Omega_{\rm formal}$. From the first part of Proposition 3.6 it follows that $\mathscr{C}_{p}(\omega)=\frac{A({\bf x})}{f^{\sigma}({\bf x})}+p\omega_{1}$ for some $\omega_{1}\in\widehat{\Omega}_{f^{\sigma}}(\mu)$ and $A({\bf x})$ a Laurent polynomial with support in $\mu_{\mathbb{Z}}$. Since $\omega\in d\Omega_{\rm formal}$ we have that the Laurent series of $\mathscr{C}_{p}(\omega)$ is divisible by $p$. This implies that $p$ divides $A({\bf x})$. Hence $\mathscr{C}_{p}(\omega)=p\omega_{2}$ with $\omega_{2}=\omega_{1}+A({\bf x})/pf^{\sigma}({\bf x})\in\widehat{\Omega}_{f^{\sigma}}(\mu)$. Applying this observation recursively then yields $\mathscr{C}_{p}^{s}(\omega)\in p^{s}\widehat{\Omega}_{f^{\sigma^{s}}}(\mu)$, which ends our proof. ∎

It is clear from Definition 4.1 that the Cartier operator preserves this submodule and is divisible by $p$ on it, that is we have

[TABLE]

Recall that the Cartier operator commutes with the connection operations for all derivations $\delta$ of $R$. It is then immediate from Definition 4.1 that all $\delta$ preserve $U_{f}(\mu)$. In other words, $U_{f}(\mu)$ is a differential submodule of $\widehat{\Omega}_{f}(\mu)$.

Theorem 4.3**.**

Assume that the Hasse–Witt matrix $\beta_{p}(\mu)$ is invertible in $R$. Then the quotient

[TABLE]

is a free $R$-module of rank $h=\#\mu_{\mathbb{Z}}$ with a basis given by the images of $\omega_{{\bf u}},{\bf u}\in\mu_{\mathbb{Z}}$.

Strictly speaking,the quotient $\widehat{\Omega}_{f}(\mu)/d\Omega_{\rm formal}$ should be read as $\widehat{\Omega}_{f}(\mu)/U_{f}(\mu)$ since $U_{f}(\mu)=\widehat{\Omega}_{f}(\mu)\cap d\Omega_{\rm formal}$. We prefer to use the former, more suggestive, notation.

Remark 4.4**.**

Recall that we work under assumptions that $\cap_{s}p^{s}R=\{0\}$ and $R$ is $p$-adically complete, in which case an element of $R$ is invertible if and only if it is invertible modulo $p$. Indeed, if $uv=1+pw$ then the inverse element is given by $u^{-1}=v(1+pw)^{-1}=\sum_{k\geq 0}(-p)^{k}vw^{k}$. With this observation, we conclude from Theorem 4.3 and Proposition 3.6 that the Cartier operator on the quotients

[TABLE]

is invertible because its matrix in the bases $\{\omega_{\bf u}\},\{\omega_{\bf u}^{\sigma}\}$ is congruent modulo $p$ to the (transposed) Hasse–Witt matrix $\beta_{p}(\mu)$.

Later we will give an explicit $p$-adic formula for the Cartier matrices on the quotients $Q_{f}(\mu)$ using matrices $\beta_{p^{s}}(\mu)$ for $s\geq 1$ (see Theorem 5.3). The proof of Theorem 4.3 exploits the $p$-adic contraction property of the Cartier operator from Proposition 3.6. The main argument is essentially contained in the following

Proposition 4.5**.**

Let $M_{0},M_{1},M_{2},\ldots$ be an infinite sequence of $R$-modules and $\phi_{i}:M_{i-1}\to M_{i}$ $R$-linear maps for all $i\geq 1$. Suppose that $\cap_{s\geq 1}p^{s}M_{i}=\{0\}$ for all $i$. For each $i$ let $N_{i}$ be a submodule of $M_{i}$ such that $\phi_{i}(M_{i-1})\subset N_{i}+pM_{i}$ for all $i\geq 1$. Suppose that $N_{i}\cap pM_{i}=pN_{i}$ (equivalently, $M_{i}/N_{i}$ is $p$-torsion free) and the induced maps $\phi_{i}:N_{i-1}/pN_{i-1}\to N_{i}/pN_{i}$ are isomorphisms for all $i\geq 1$. Define submodules

[TABLE]

Then, for all $i$,

1. (i)

$M_{i}=N_{i}+U_{i}$. 2. (ii)

$\phi_{i}(U_{i-1})\subset pU_{i}$. 3. (iii)

$\phi_{i}(M_{i-1})\subset N_{i}+pU_{i}$. 4. (iv)

$N_{i}\cap U_{i}=\{0\}$.

Proof.

Note that (ii) is an immediate consequence of the definition of $U_{i}$’s. Indeed, for $\omega\in U_{i-1}$ the element $\omega_{1}\in M_{i}$ such that $\phi_{i}(\omega)=p\omega_{1}$ satisfies $\phi_{i+s}\circ\cdots\circ\phi_{i+1}(\omega_{1})=\frac{1}{p}\phi_{i+s}\circ\cdots\circ\phi_{i}(\omega)\in p^{s}M_{i+s}$ for all $s\geq 1$.

Let us show that (iii) follows easily from (i). For any $\omega\in M_{i-1}$ write $\phi_{i}(\omega)=\omega_{1}+p\omega_{1}^{\prime}$ with $\omega_{1}\in N_{i}$, $\omega_{1}^{\prime}\in M_{i}$. Using (i) we can write $\omega_{1}^{\prime}=\nu_{1}+u_{1}$ with $\nu_{1}\in N_{i}$, $u_{1}\in U_{i}$. Thus we get $\phi_{i}(\omega)=\omega_{1}+\nu_{1}+pu_{1}\in N_{i}+pU_{i}$.

Proof of (i). Clearly it is enough to do it for $i=0$.

Consider $\phi_{i}$ modulo $p$, which is a map from $M_{i-1}/pM_{i-1}$ to $M_{i}/pM_{i}$. By the assumption that $\phi_{i}(M_{i-1})\subset N_{i}+pM_{i}$, the image of $\phi_{i}({\rm mod}\ p)$ lies in $N_{i}/(N_{i}\cap pM_{i})$. Since we also assume that $N_{i}\cap pM_{i}=pN_{i}$, the image of $\phi_{i}({\rm mod}\ p)$ lies in $N_{i}/pN_{i}$. Restricting $\phi_{i}({\rm mod}\ p)$ to $N_{i-1}/pN_{i-1}$ we obtain what we call the induced map $\phi_{i}:N_{i-1}/pN_{i-1}\to N_{i}/pN_{i}$. It is assumed that this induced map is invertible for each $i$, and hence the composition $\psi_{i}:=\phi_{i}\circ\cdots\phi_{2}\circ\phi_{1}:N_{0}/pN_{0}\to N_{i}/pN_{i}$ is an isomorphism for all $i\geq 1$. Define for each $s\geq 1$

[TABLE]

and $U_{0}^{(0)}=M_{0}$. In particular observe that $U_{0}^{(s+1)}\subset U_{0}^{(s)}$ for all $s\geq 0$. We first show that $U_{0}^{(s)}=U_{0}^{(s+1)}+p^{s}N_{0}$ for all $s\geq 0$. Let $\omega\in U_{0}^{(s)}$. Then $\omega_{s}:=p^{-s}\psi_{s}(\omega)\in M_{s}$. By our assumptions there exists $\eta_{s}\in N_{s}$ such that $\phi_{s+1}(\eta_{s})\equiv\phi_{s+1}(\omega_{s})({\rm mod}\ pM_{s+1})$. Choose $\eta_{0}\in N_{0}$ such that $\psi_{s}(\eta_{0})\equiv\eta_{s}({\rm mod}\ pM_{s})$. Then,

[TABLE]

Hence $\omega-p^{s}\eta_{0}\in U_{0}^{(s+1)}$.

Let $\omega\in M_{0}$. For $s\geq 1$ we define $\omega_{s}\in U_{0}^{(s)}$ inductively via $\omega_{s}=\omega_{s+1}+p^{s}\eta_{s},\ \eta_{s}\in N_{0}$. One easily sees that $\omega-\sum_{s\geq 1}p^{s}\eta_{s}\in\cap_{s\geq 1}U_{0}^{(s)}=U_{0}$. Hence we conclude that $M_{0}=N_{0}+U_{0}$.

We finally show that $N_{0}\cap U_{0}$ is trivial. Suppose, on the contrary, that $\omega\in N_{0}\cap U_{0}$ and $\omega\neq 0$. Because $\cap_{s\geq 1}p^{s}M_{0}=\{0\}$ there exists $s\geq 0$ such that $p^{-s}\omega\in M_{0}\setminus pM_{0}$. Since $M_{0}/N_{0}$ is $p$-torsion free this implies that $p^{-s}\omega\in N_{0}\setminus pM_{0}$. Since $\psi_{i}:N_{0}/pN_{0}\to N_{i}/pN_{i}$ is an isomorphism we have that $\psi_{i}(p^{-s}\omega)\not\in pM_{i}$ for all $i$. In particular for $i=s+1$ we get $\psi_{s+1}(p^{-s}\omega)\not\in pM_{s+1}$. Hence $\psi_{s+1}(\omega)\not\in p^{s+1}M_{s+1}$. This contradicts the fact that $\omega\in U_{0}$. Thus we get a contradiction and conclude that $N_{0}\cap U_{0}$ is trivial. ∎

Proof of Theorem 4.3..

We apply Proposition 4.5 to $M_{i}=\widehat{\Omega}_{f^{\sigma^{i}}}(\mu)$ and $\phi_{i}=\mathscr{C}_{p}$ for all $i\geq 0$. For $N_{i}$ we take the ${\rm Span}_{R}(\omega^{\sigma^{i}}_{{\bf u}})_{{\bf u}\in\mu_{\mathbb{Z}}}$. The property $N_{i}\cap pM_{i}=pN_{i}$ clearly holds. Proposition 3.6 states that $\phi_{i}(M_{i})\subset N_{i}+pM_{i}$ and the matrix of $\phi_{i}:N_{i-1}/pN_{i-1}\to N_{i}/pN_{i}$ is given by $\beta_{p}^{\sigma^{i-1}}(\mu)$ modulo $p$, which is invertible by the assumption in Theorem 4.3. So the assumptions of Proposition 4.5 are satisfied.

From Proposition 4.2 we find that $U_{0}=U_{f}(\mu)$. Then application of parts (i) and (iv) of Proposition 4.5 shows that

[TABLE]

as $R$-modules. Hence $Q_{f}\cong{\rm Span}_{R}(\omega_{{\bf u}})_{{\bf u}\in\mu_{\mathbb{Z}}}$. ∎

Remark 4.6**.**

Parts (iii) and (iv) in Proposition 4.5 imply that

[TABLE]

Remark 4.7**.**

Theorem 4.3 is not true if we would have defined $\omega_{{\bf u}}$ without the factorial $(u_{0}-1)!$. To see this take the simplest example $f=1-x$ in one variable and $R={\mathbb{Z}}_{p}$. Theorem 4.3 implies that every rational function $(k-1)!\frac{x^{r}}{(1-x)^{k}}$ with $0\leq r\leq k$ is modulo (formal) derivatives equivalent to a function of the form $A+\frac{B}{1-x}$. Now drop the factorial, take $k=p+1,r=0$ and suppose there exist $A,B\in{\mathbb{Z}}_{p}$ such that $A+\frac{B}{1-x}=\frac{1}{(1-x)^{p+1}}+xu^{\prime}$ for some rational function $u$. Apply $\mathscr{C}_{p}$ modulo $p$ on both sides. The derivative $xu^{\prime}$ is mapped to [math], $\frac{1}{1-x}$ is mapped to itself and we get

[TABLE]

On the right of this equality we see a rational function with a double pole at $x=1$ on the left a simple pole. This is clearly contradictory.

Remark 4.8**.**

Knowledge of the explicit basis in $Q_{f}(\mu)$ from Theorem 4.3 implies that this $R$-module is in fact a quotient of $\Omega_{f}(\mu)$. However writing it as a quotient of the completion $\widehat{\Omega}_{f}(\mu)$ yields the Cartier operator on $Q_{f}(\mu)$. Note also that $Q_{f}(\mu)$ is a subquotient of the Dwork module $\mathscr{W}_{f}$ because derivatives are contained in $U_{f}(\mu)$.

We would like to point out that $R$-modules $\widehat{\Omega}_{f}(\mu)$, $U_{f}(\mu)$, completed Dwork modules $\widehat{\Omega}_{f}(\mu)/d\widehat{\Omega}_{f}(\mu)$ and the quotients $Q_{f}(\mu)$ from Theorem 4.3 together with the Cartier operator $\mathscr{C}_{p}$ are examples of the following structure. For the scope of this paper, we give the following

Definition 4.9**.**

A crystal over $R$ is a rule that assigns

• •

to a polynomial $f$ with coefficients in $R$ a differential $R$-module $M_{f}$, that is for every derivation $\delta$ of $R$ we have maps $\delta:M_{f}\to M_{f}$ satisfying $\delta(rm)=\delta(r)m+r\delta(m)$ for $r\in R,m\in M_{f}$ (connection maps);

• •

*to every *$p$th power Frobenius lift $\sigma:R\to R$ an $R$-linear map $\mathscr{C}_{p}:M_{f}\to M_{f^{\sigma}}$ which commutes with the connection, that is we have $\mathscr{C}_{p}\circ\delta=\delta\circ\mathscr{C}_{p}$ for every derivation $\delta$ of $R$.

Note that over rings $R$ which have no non-trivial derivations, e.g. ${\mathbb{Z}}_{p}$ and its finite extensions, it still makes sense to consider crystals, though the conditions related to connection are empty.

Following the traditional terminology, see e.g. [8], one can call $Q_{f}(\mu)$ the unit-root quotient in reflection of the fact that the Cartier operator is divisible by $p$ on $U_{f}(\mu)$ and invertible on the quotient $Q_{f}(\mu)=\widehat{\Omega}_{f}(\mu)/U_{f}(\mu)$ (see Remark 4.4).

Note also that, when the Hasse–Witt matrix is invertible, $U_{f}(\mu)\subset\widehat{\Omega}_{f}(\mu)$ can be characterized as the largest subcrystal on which the Cartier operator is divisible by $p$.

5. Periods mod $m$

For any exponent vector ${\bf v}\in C(\Delta)_{\mathbb{Z}}$ we define the linear functional $\tau_{{\bf v}}$ on $\Omega_{\rm formal}$ by

[TABLE]

Lemma 5.1**.**

Let $m\geq 1$ be an integer. For any $\omega\in d\Omega_{\rm formal}$ we have $\tau_{m{\bf v}}(\omega)\equiv 0({\rm mod}\ m)$.

For any $\omega\in\Omega_{\rm formal}$ and any derivation $\delta$ of $R$ we have $\delta(\tau_{m{\bf v}}(\omega))\equiv\tau_{m{\bf v}}(\delta(\omega))({\rm mod}\ m)$.

Proof.

Suppose that $\omega=x_{i}\frac{\partial u}{\partial x_{i}}$ for some Laurent expansion $u$. Then

[TABLE]

For any derivation $\delta$ of $R$ and any Laurent series $\omega$ we have

[TABLE]

because operations $\delta$ and taking the constant term commute and derivation of an $m$th power is zero modulo $m$. ∎

The two properties in Lemma 5.1 show that functionals $\tau_{m{\bf v}}$ restricted modulo $m$ are what we call period maps modulo $m$. That is, they are $R$-linear maps from $\Omega_{\rm formal}$ to $R/mR$ that vanish on derivatives and commute with derivations of $R$.

Next, we look at the behaviour of these linear functionals under the Cartier operator:

Proposition 5.2**.**

Let $p$ be a prime and $\sigma:R\to R$ be a $p$th power Frobenius lift. Denote by $\tau^{\sigma}_{m{\bf v}}$ the linear functional obtained by multiplication with $(f^{\sigma})^{mv_{0}}/{\bf x}^{m{\bf v}}$ and then taking the constant term. Then

[TABLE]

Proof.

For any $\omega\in\Omega_{\rm formal}$ we have

[TABLE]

The second step uses the obvious fact that the constant term equals the constant term of the Cartier transform. In the last step we used a variant of Lemma 3.2 in the bigger ring $R[{\bf x}]\otimes_{R}\Omega_{\rm formal}$. ∎

The period maps introduced here are useful when working in $\Omega_{f}$. Note that to compute $\tau_{{\bf v}}(\omega_{\bf u})$ we simply take the constant coefficient of the product $\frac{f^{v_{0}}}{{\bf x}^{{\bf v}}}\omega_{{\bf u}}$, which is a Laurent polynomial when $u_{0}\leq v_{0}$. In the particular case when $u_{0}=v_{0}=1$ we observe that $\tau_{m{\bf v}}(\omega_{{\bf u}})=(\beta_{m})_{{\bf u},{\bf v}}$ for each $m\geq 1$, where $\beta_{m}$ are the matrices defined in (8).

The following theorem is our second main result.

Theorem 5.3**.**

Let $\mu\subseteq\Delta$ be a set open in the topology defined in Proposition 3.4. Suppose that $R$ is $p$-adically complete and the Hasse–Witt matrix $\beta_{p}(\mu)$ is invertible in $R$. Then $\beta_{p^{s}}(\mu)$ is invertible for all $s\geq 1$.

Let $Q_{f}(\mu)$ be the unit-root crystal from Theorem 4.3 and let $\Lambda_{\sigma}=(\lambda_{{\bf u},{\bf w}})$ be the transposed matrix of $\mathscr{C}_{p}:Q_{f}(\mu)\to Q_{f^{\sigma}}(\mu)$ with respect to the standard bases $\{\omega_{{\bf u}}\},\{\omega^{\sigma}_{{\bf w}}\}$. More precisely, it is the $h\times h$-matrix with entries in $R$ such that

[TABLE]

for all ${\bf u}\in\mu_{\mathbb{Z}}$. Then, for all $s\geq 1$ and all $m\geq 1$, it satisfies the congruences

[TABLE]

In particular, when $m=1$,

[TABLE]

Similarly, for every derivation $\delta$ of $R$ Theorem 4.3 implies that there exists a unique matrix $N_{\delta}=(\nu_{{\bf u},{\bf v}})_{{\bf u},{\bf v}\in\mu_{\mathbb{Z}}}$ with entries in $R$ such that

[TABLE]

This matrix then satisfies congruences

[TABLE]

for all $m,s\geq 1$. In particular, when $m=1$,

[TABLE]

Proof.

Using (iii) in Proposition 4.5, the congruence (10) can be refined to

[TABLE]

(see Remark 4.6). We apply $\tau^{\sigma}_{mp^{s-1}{\bf v}}$ with ${\bf v}\in\mu_{\mathbb{Z}}$ to (14). By Proposition 5.2 we have

[TABLE]

in the left-hand side. Since elements of $U_{f^{\sigma}}(\mu)$ are formal derivatives (see Proposition 4.2), in the right-hand side Lemma 5.1 yields $p\,\tau^{\sigma}_{mp^{s-1}{\bf v}}(U_{f^{\sigma}}(\mu))\equiv 0({\rm mod}\ p^{s})$. So we obtain congruence

[TABLE]

It follows that $\beta_{mp^{s}}(\mu)\equiv\Lambda_{\sigma}\beta^{\sigma}_{mp^{s-1}}(\mu)({\rm mod}\ p^{s})$. By Proposition 3.6, $\Lambda_{\sigma}\equiv\beta_{p}(\mu)({\rm mod}\ p)$ and we find that $\beta_{p^{s}}(\mu)\equiv\beta_{p}(\mu)\beta^{\sigma}_{p^{s-1}}(\mu)({\rm mod}\ p)$. By iteration then obtain

[TABLE]

Hence invertibility of all $\beta_{p^{s}}(\mu)$ modulo $p$ follows from the case $s=1$. After inversion of $\beta^{\sigma}_{p^{s-1}}(\mu)$ (it is invertible over $R$, see Remark 4.4) we find that $\Lambda_{\sigma}\equiv\beta_{p^{s}}(\mu)\,\beta^{\sigma}_{p^{s-1}}(\mu)^{-1}\;({\rm mod}\ p^{s})$.

The proof of the second congruence runs similarly: we apply $\tau_{mp^{s}{\bf v}}$ with ${\bf v}\in\mu_{\mathbb{Z}}$ to (13). Since $\tau_{mp^{s}{\bf v}}(U_{f}(\mu))\equiv 0\,({\rm mod}\ p^{s})$ and $\delta$ commutes with $\tau_{mp^{s}{\bf v}}$ modulo $p^{s}$ (see Lemma 5.1) we obtain

[TABLE]

Hence we conclude that $\delta(\beta_{mp^{s}}(\mu))\equiv N_{\delta}\,\beta_{mp^{s}}(\mu)({\rm mod}\ p^{s})$, as desired. ∎

Remark 5.4**.**

In [11, §1] the second author conjectured vaguely that the $p$-adic limits

[TABLE]

describe respectively the Frobenius operator and the Gauss–Manin connection on the unit-root crystal attached to the Laurent polynomial $f({\bf x})$. However the precise meaning of the unit-root crystal in the conjecture was not specified. Moreover, it looked challenging to define this object using as little assumptions on $f({\bf x})$ as one needs for existence of the $p$-adic limiting matrices (15). Theorem 5.3 implies that this conjecture is true with the unit-root crystal being the dual $Q_{f}^{\vee}={\rm Hom}_{R}(Q_{f}(\Delta^{\circ}),R)$ of the crystal defined in Theorem 4.3 with the Frobenius operator $\mathscr{C}_{p}^{\vee}:Q_{f^{\sigma}}^{\vee}\to Q_{f}^{\vee}$. Note that in addition to the invertibility of the Hasse–Witt matrix, which is needed to define (15), we only use one extra assumption: there is a vertex ${\bf b}$ of $\Delta$ such that the coefficient of $f({\bf x})$ at ${\bf b}$ is a unit in $R$. The latter is a technical assumption that was made in Section 2 for the purposes of doing formal expansion at ${\bf b}$ with integral coefficients; it is most likely that one could drop this condition as the Cartier operator (7) can be defined directly by formulas (4) and (5).

A different proof of the conjecture was given recently in [6, §5] under certain geometric assumptions.

Example 5.5**.**

Consider $f(x,y)=y^{2}-x(x-1)(x-z)\in R[x,y]$ as a polynomial with coefficients in a ring $R$ containing ${\mathbb{Z}}[z]$, which we will specify in a moment. We would like to apply Theorem 5.3 with $\mu=\Delta^{\circ}\subset{\mathbb{R}}^{2}$, the interior of the Newton polytope of $f(x,y)$. In this case $\mu_{\mathbb{Z}}=\{(1,1)\}$, $h=\#\mu_{\mathbb{Z}}=1$ and we have

[TABLE]

To shorten our notation, we will write $\beta_{m}(\mu)$ simply as $\beta_{m}$ throughout this Example. Now fix a prime $p>2$. Let $R={\mathbb{Z}}[z,\beta_{p}^{-1}]\;{\widehat{}}\subset{\mathbb{Z}}_{p}\llbracket z\rrbracket$ be the $p$-adic completion of ${\mathbb{Z}}[z,\beta_{p}^{-1}]$. This ring consists of power series $g(z)\in{\mathbb{Z}}_{p}\llbracket z\rrbracket$ that can be approximated $p$-adically by rational functions whose denominators are powers of the Hasse–Witt polynomial $\beta_{p}\in{\mathbb{Z}}[z]$ in the denominator. One can check that the Frobenius lift $\sigma$ given by $(\sigma g)(z)=g(z^{p})$ preserves $R$. We claim that the respective Cartier matrix (10), which is now a $1\times 1$-matrix, is given by

[TABLE]

where

[TABLE]

is the hypergeometric series mentioned in the Introduction. Note in particular, that this statement implies that $F(z)/F(z^{p})\in{\mathbb{Z}}[z,\beta_{p}^{-1}]\;{\widehat{}}$.

To prove (16) we notice that $\frac{(1/2)_{k}}{k!}=\frac{\Gamma(k+\frac{1}{2})}{\Gamma(k+1)\,\Gamma(\frac{1}{2})}=\frac{(-1)^{k}\Gamma(\frac{1}{2})}{\Gamma(k+1)\,\Gamma(\frac{1}{2}-k)}=(-1)^{k}\binom{-\frac{1}{2}}{k}$ and

[TABLE]

The latter congruence can be checked by induction on $k$. Since $\mathrm{ord}_{p}(k!)\leq\frac{k}{(p-1)}$, it follows that

[TABLE]

This congruence is much weaker than the one in (12). However it is sufficient to conclude that the $p$-adic limit $\Lambda_{\sigma}=\lim_{s\to\infty}\beta_{p^{s}}/\sigma(\beta_{p^{s-1}})$ equals $F(z)/F(z^{p})$ times the $p$-adic limit of the ratios $\binom{p^{s}-1}{(p^{s}-1)/2}/\binom{p^{s-1}-1}{(p^{s-1}-1)/2}$. One can check that such a ratio is congruent to $(-1)^{\frac{p-1}{2}}$ modulo $p^{s}$, which completes our proof of (16). In a similar vein, one can show that $N_{\delta}=(\delta{F})(z)/F(z)$ for a derivation $\delta$ of $R$.

Let us mention an application of congruence (11) to integrality of formal group laws. Consider a $h$-tuple of formal powers series $l(z)=(l_{\bf u}(z))_{{\bf u}\in\mu_{\mathbb{Z}}}$ in $h$ variables $z=(z_{\bf v})_{{\bf v}\in\mu_{\mathbb{Z}}}$ given by

[TABLE]

These power series have coefficients in $R\otimes{\mathbb{Q}}$ and satisfy $l_{\bf u}(z)\equiv z_{\bf u}$ modulo terms of degree $\geq 2$.

Corollary 5.6**.**

Under the assumptions of Theorem 5.3, the $h$-dimensional formal group law

[TABLE]

has coefficients in $R$.

Proof.

Since $R$ is a ${\mathbb{Z}}_{p}$-algebra, congruences (11) are equivalent to the statement that the tuple of power series $l(z)-p^{-1}\Lambda_{\sigma}\,l^{\sigma}(z^{p})$ has coefficients in $R$. Integrality of $G(z,z^{\prime})$ then follows from Hazewinkel’s functional equation lemma [5, §10.2]. ∎

Formal group laws $G(z,z^{\prime})$ in Corollary 5.6 include coordinalizations of some Artin–Mazur formal groups of algebraic varieties, see [10, Theorem 1]. In the very particular example $f=y^{2}-x(x-1)(x-z)$ from the introduction with $\mu=\Delta^{\circ}$ it follows from [10, p. 924] that the formal group is simply the formal law of addition on the elliptic curve $f=0$.

Now we would like to explain the connection between our results and [8]. For that purpose, consider linear functionals on $\Omega_{\rm formal}$ given by

[TABLE]

for {\bf k}\in C(\Delta{\color[rgb]{0,0,1}-}{\bf b})_{\mathbb{Z}}. Just as we had above with $\tau_{{\bf v}}$, for any ${\bf k}$ and $m\geq 1$ functional $\alpha_{m{\bf k}}$ is a period modulo $m$. Indeed, by Lemma 2.2 this functional takes values in $mR$ on formal derivatives and, since derivations of $R$ act on $\Omega_{\rm formal}$ simply by applying them to coefficients, we clearly have $\alpha_{m{\bf k}}\circ\delta=\delta\circ\alpha_{m{\bf k}}$. These periods have an obvious property with respect to the Cartier operator:

[TABLE]

for all $m$ divisible by $p$. Combining these observations with Theorem 4.3 we obtain the following version of [8, Theorem 6.2]:

Theorem 5.7**.**

Let $\mu\subseteq\Delta$ be a set open in the topology defined in Proposition 3.4 and $h=\#\mu_{\mathbb{Z}}$. For {\bf k}\in C(\Delta{\color[rgb]{0,0,1}-}{\bf b})_{\mathbb{Z}} consider the column vector ${\bf a}_{{\bf k}}\in R^{h}$ with components

[TABLE]

Assume that $R$ is $p$-adically complete and the Hasse–Witt matrix $\beta_{p}(\mu)$ is invertible in $R$. For any Frobenius lift $\sigma$ and any derivation $\delta$ of $R$, let $\Lambda_{\sigma}$ and $N_{\delta}$ $\Lambda_{\sigma},N_{\delta}\in R^{h\times h}$ be the matrices defined in (10) and (13) respectively. (These matrices correspond to the Cartier operator and connection on the unit-root crystal $Q_{f}(\mu)$ defined in Theorem 4.3.) We then have

[TABLE]

and

[TABLE]

for all ${\bf k}\in C(\Delta-{\bf b})_{\mathbb{Z}}$.

Proof.

Consider the equality

[TABLE]

Expand all terms in a Laurent series with respect to the vertex ${\bf b}$ and determine the coefficient of ${\bf x}^{{\bf k}p^{s-1}}$ on both sides. For the term in $pU^{\sigma}_{f}$ we get a value $0({\rm mod}\ p^{s})$. The other terms give us

[TABLE]

which gives us the first statement.

For the second statement we start with

[TABLE]

Expand as Laurent series and take the coefficient of ${\bf x}^{{\bf k}p^{s}}$ on both sides. We get

[TABLE]

which proves our second statement. ∎

We end with an application of Theorems 5.3 and 5.7.

Corollary 5.8**.**

Suppose that $\mu$ is an open set that consists of one vertex point ${\bf v}\in\Delta$. Let $f_{{\bf v}}$ be the coefficient of ${\bf x}^{{\bf v}}$ in $f$ and suppose it is a unit in $R$. Then we have the equality $\mathscr{C}_{p}(\omega_{{\bf v}})\equiv\frac{f_{{\bf v}}^{\sigma}}{f_{{\bf v}}}\omega_{{\bf v}}^{\sigma}({\rm mod}\ U_{f^{\sigma}}(\mu))$.

Proof.

This follows almost immediately from Theorem 5.3. Note that $\beta_{p^{s}}(\mu)$ is a $1\times 1$-matrix with entry $f_{{\bf b}}^{p^{s}-1}$. The matrix $\Lambda_{\sigma}$ has the entry $\lim_{s\to\infty}f_{{\bf b}}^{p^{s}-1}/(f_{{\bf b}}^{\sigma})^{p^{s-1}-1}=f_{{\bf b}}^{\sigma}/f_{{\bf b}}$. ∎

Note that the situation when one vertex is an open set in the topology from Proposition 3.4 can occur if all lattice points in $\Delta$ are vertices. The complement of all but one of these vertices gives us an open one-point set $\mu$.

The following corollary is a generalization of Theorem 5.6 in [2], which deals with congruences for coefficients of power series expansions of rational functions.

Corollary 5.9**.**

Let $f({\bf x})$ be a Laurent polynomial with coefficients in ${\mathbb{Z}}_{p}$ such that all lattice points in its Newton polytope $\Delta\subset{\mathbb{R}}^{n}$ are vertices. Suppose that all coefficients of $f({\bf x})$ are $p$-adic units. Let $g({\bf x})$ be a Laurent polynomial with coefficients in ${\mathbb{Z}}_{p}$ and support in $\Delta$. Choose any vertex ${\bf b}\in\Delta$ and consider the respective formal expansion

[TABLE]

Then, for every ${\bf k}\in C(\Delta-{\bf b})$ and $s\geq 1$ we have $a_{p^{s}{\bf k}}\equiv a_{p^{s-1}{\bf k}}({\rm mod}\ p^{s})$.

Proof.

It is sufficient to give a proof for a monomial $g({\bf x})={\bf x}^{\bf v}$, ${\bf v}\in\Delta_{\mathbb{Z}}$. Application of Theorem 5.7 with $\mu=\{{\bf v}\}$, which is an open set due to our assumption on $\Delta$, yields the congruence $a_{p^{s}{\bf k}}\equiv\Lambda_{\sigma}a_{p^{s-1}{\bf k}}({\rm mod}\ p^{s})$ with $\Lambda_{\sigma}\in{\mathbb{Z}}_{p}^{\times}$. Since $R={\mathbb{Z}}_{p}$ we have that $f^{\sigma}_{{\bf v}}=f_{{\bf v}}$ and hence $\Lambda_{\sigma}=1$ as in Corollary 5.8. ∎

In [2] the polytope $\Delta$ is a subset of the unit hypercube in ${\mathbb{R}}^{n}$, hence the conditions of Corollary 5.9 are satisfied.

6. Semi-simple decomposition

Let $\tilde{n}\leq n$ be the dimension of $\Delta$. For $0\leq l\leq\tilde{n}$ let $\mu^{(l)}\subset\Delta$ be the complement of the union of faces of codimension $>l$; this is an open set in the topology defined in Proposition 3.4. The inclusions

[TABLE]

give rise to a filtration on the module of regular functions given by

[TABLE]

Note that this filtration is preserved by the connection and its $p$-adic completion is preserved by the Cartier operator, i.e. $\mathscr{C}_{p}:\widehat{\Omega}_{f}^{n}(\mu^{(l)})\to\widehat{\Omega}_{f^{\sigma}}^{n}(\mu^{(l)})$ for each $l$ (see Proposition 3.4). We quotient the $p$-adic completions by formally exact forms and obtain

[TABLE]

Let $\beta_{p}^{(l)}$ be the Hasse–Witt matrix of $f$ relative to $\mu^{(l)}$. We shall call $\beta_{p}^{(0)}$ simply the Hasse–Witt matrix of $f$. The following fact is a straightforward corollary of the congruences stated in Theorem 5.3. As in this theorem, we assume that $\cap_{s\geq 0}p^{s}R=\{0\}$ and $R$ is $p$-adically complete. Recall that in this case an element is invertible if and only if it is invertible modulo $p$. Note also that for any face $\eta\subset\Delta$ the Newton polytope of the restriction $f|_{\eta}$ is given by $\eta$.

Theorem 6.1**.**

Assume that the coefficients of $f$ at all vertices of $\Delta$ are units in $R$. Let $l\geq 1$. The matrix $\beta_{p}^{(l)}$ is invertible if and only if $\beta_{p}^{(l-1)}$ and the Hasse–Witt matrices of all restrictions $f|_{\eta}$ to the faces $\eta\subset\Delta$ of codimension $l$ are invertible. If this is the case, one has the following decomposition of the quotient crystal

[TABLE]

where $\eta^{\circ}$ is the interior of the face $\eta$, and we make the convention that an interior of a vertex is the vertex itself.

Proof.

We write $\mu^{(\ell)}=\eta_{1}^{\circ}\cup\eta_{2}^{\circ}\cup\ldots\cup\mu^{(\ell-1)}$, where $\eta_{1},\eta_{2},\ldots$ are all faces of $\Delta$ of codimension $\ell$, and claim that for any $m\geq 1$ matrices $\beta_{m}(\mu^{(\ell)})$ have the following block structure

[TABLE]

with diagonal blocks corresponding to all faces of codimension $\ell$ and $\mu^{(\ell-1)}$ and possibly non-zero off-diagonal blocks only in the last column. This claim follows from the following observation: if $\eta\subsetneq\Delta$ is a face, ${\bf v}\in\eta$ and ${\bf u}\not\in\eta$, then the coefficients of $m{\bf v}-{\bf u}$ in $f({\bf x})^{m-1}$ is zero. Indeed, choose a linear functional $\kappa:{\mathbb{R}}^{n}\to{\mathbb{R}}$ such that $\kappa|_{\eta}\equiv c$, and $\kappa(\Delta)\subset{\mathbb{R}}_{\leq c}$ and $\kappa({\bf u})<c$ for some $c\in{\mathbb{R}}$. Then $\kappa(m{\bf v}-{\bf u})=mc-\kappa(u)>(m-1)c$ and therefore $m{\bf v}-{\bf u}\not\in(m-1)\Delta$.

Taking $m=p$ we see that the Hasse–Witt matrix $\beta_{p}(\mu^{\ell})$ is invertible if and only if all $\beta_{p}(\eta_{i}^{\circ})$ and $\beta_{p}(\mu^{\ell-1})$ are invertible. Since the above mentioned block structure is preserved under taking the inverse, by the congruences in Theorem 5.3 matrices $\Lambda_{\sigma}$ and $N_{\delta}$ for $\mu^{(\ell)}$ have the same block structure and the direct sum decomposition of the quotient crystal follows immediately.

∎

Remark 6.2**.**

Corollaries 5.8 and 5.9 deal with the situation when the only lattice points in the Newton polytope $\Delta$ are its vertices. In this case filtration (20) has only one step (the set $\mu^{(\tilde{n}-1)}_{\mathbb{Z}}$ is empty) and, assuming that the coefficients of $f({\bf x})=\sum_{{\bf v}}f_{\bf v}{\bf x}^{\bf v}$ at all vertices are units in $R$, Theorem 6.1 states that the unit-root crystal $Q_{f}=Q_{f}(\Delta)$ is a direct sum of crystals of rank 1.

Let us mention that in the regular case (i.e. when $f$ is $\Delta$-regular) under the identification of the Dwork module $\mathscr{W}_{f}=\Omega_{f}/d\Omega_{f}$ with the cohomology group $H^{n}({\mathbb{T}}^{n}\setminus Z_{f})$ (after tensoring with the field of fractions of $R$) the image of the filtration (20) is the weight filtration of the respective mixed Hodge structure (see [1, Theorem 8.2]). Theorem 6.1 thus gives a semi-simple decomposition of unit-root crystals corresponding to the graded pieces of the weight filtration.

7. Example

Consider $f=y^{2}+tx^{3}+xy+x$ and $R$ the $p$-adic completion of ${\mathbb{Z}}_{p}[t]$. We have the following sets of exponent vectors with $u_{0}=1$,

1. (1)

$\mu^{(2)}_{\mathbb{Z}}=\{(0,2),(1,0),(3,0),(2,0),(1,1)\}=\Delta_{\mathbb{Z}}$ 2. (2)

$\mu^{(1)}_{\mathbb{Z}}=\{(2,0),(1,1)\}$ 3. (3)

$\mu^{(0)}_{\mathbb{Z}}=\{(1,1)\}$

where the $\mu^{(l)}$ are defined in the previous section. The ordering of the exponent vectors in $\Delta$ is chosen in decreasing filtration order. Using this ordered basis a straightforward calculation shows that for odd $m$,

[TABLE]

where

[TABLE]

[TABLE]

For the invertibility of $\beta_{p}$ we extend $R$ to be the $p$-adic completion of ${\mathbb{Z}}_{p}[t,(tf_{p}(t))^{-1}]$. Then $\beta_{p^{s}}(\beta_{p^{s-1}}^{\sigma})^{-1}$ reads

[TABLE]

where $c(p,s)={p^{s}-1\choose(p^{s}-1)/2}{p^{s-1}-1\choose(p^{s-1}-1)/2}^{-1}$. We now take the limit as $s\to\infty$. It is not hard to derive that $c(p,s)\equiv(-1)^{(p-1)/2}({\rm mod}\ p^{s})$. Also, using ${p^{s}-1\choose m}\equiv{p^{s-1}-1\choose\lfloor m/p\rfloor}({\rm mod}\ p^{s})$, one easily shows that

[TABLE]

Finally, experiment shows that

[TABLE]

where $F(t)=\,_{2}F_{1}(1/4,3/4,1|64t)$ and

[TABLE]

where $G(t)=\,_{3}F_{2}(5/4,3/4,1/2;3/2,1|64t)$. Putting everything together we find the limit

[TABLE]

We make a few observations.

• (i)

$(1,1,t,0,1)\Lambda=(1,1,t^{p},0,1)$

• (ii)

$(2,0,0,0,1)\Lambda=(2,0,0,0,1)$

• (iii)

$(0,1,3t,0,1)\Lambda=(0,1,3t^{p},0,1).$

These equalities imply that in $Q_{f}$ we have

[TABLE]

where $f_{0}=f,f_{1}=x\frac{\partial f}{\partial x},f_{2}=y\frac{\partial f}{\partial y}$. This is a general phenomenon, as shown in the following theorem.

Proposition 7.1**.**

Let $f({\bf x})=f(x_{1},\ldots,x_{n})$ be a Laurent polynomial with coefficients in a characteristic zero ring $R$ such that at least one vertex coefficient of $f$ is a unit modulo $p$. Let $f_{0}=f$ and $f_{i}=x_{i}\frac{\partial f}{\partial x_{i}}$ for $i=1,\ldots,n$. Then for all $i$,

[TABLE]

Proof.

The case $i=0$ comes down to $\mathscr{C}_{p}(1)=1$, which is trivial. So let $i>0$. As earlier, we will use the notation $\theta_{i}=x_{i}\frac{\partial}{\partial x_{i}}$. Notice that

[TABLE]

where $pG({\bf x})=f({\bf x})^{p}-f^{\sigma}({\bf x}^{p})$. Observe that $\mathscr{C}_{p}(\log f^{\sigma}({\bf x}^{p}))=\log f^{\sigma}({\bf x})$. Power series expansion of the $\log$ in

[TABLE]

gives us

[TABLE]

Combining these evaluations gives us the final result

[TABLE]

Clearly the latter summation belongs to $p\,d\widehat{\Omega}_{f^{\sigma}}$ when $p>2$. ∎

We now determine the limit of

[TABLE]

where $\theta=t\frac{d}{dt}$. Some experiment suggests the following congruences

[TABLE]

and

[TABLE]

This yields the limit matrix

[TABLE]

From this limit matrix it easily follows that $\frac{f_{i}}{f},i=0,1,2$ are horizontal in $Q_{f}$, that is they are annihilated by $\theta$. This is a general phenomenon.

Proposition 7.2**.**

Let notations be as in Proposition 7.1 and $\delta$ be a derivation on $R$. Then we have

[TABLE]

Proof.

The proof is immediate,

[TABLE]

∎

Finally, getting back to our example, we mention the matrix

[TABLE]

This is a fundamental solution matrix of the system of first order equations

[TABLE]

where ${\bf y}$ is a column vector of $5$ unknown functions in $t$.

Note that Propositions 7.1 and 7.2 have nothing to do with the unit-root crystal $Q_{f}$: their statements hold modulo $d\widehat{\Omega}_{f^{\sigma}}$ and $d\Omega_{f}$ respectively and not just modulo formal derivatives. These propositions show that $\oplus_{i=0}^{n}R\frac{f_{i}}{f}$ is a subcrystal in the completed Dwork crystal $\widehat{\mathscr{W}}_{f}=\widehat{\Omega}_{f}/d\widehat{\Omega}_{f}$, on which the Cartier operator acts as the identity. In the geometric situation mentioned at the end of the introductory section, this subcrystal should correspond to the embedding of $H^{n}_{dR}({\mathbb{T}}^{n})$ into $H^{n}_{dR}({\mathbb{T}}^{n}\setminus Z_{f})\cong\mathscr{W}_{f}$.

Acknowledgement

The authors are grateful to Alan Adolphson for his comments on the manuscript.

We would like to thank to the MATRIX institute in Creswick, Max Planck Institute and Hausdorff Research Institute for Mathematics in Bonn, the University of Utrecht and Institute of Mathematics of the Polish Academy of Sciences in Warsaw, where collaboration on this project partly took place. The second author also thanks to the Mathematical Sciences Research Institute in Berkeley.

Appendix A Point counting and an alternative construction of the Cartier operator

Suppose that $R={\mathbb{Z}}_{q}$ where $q=p^{a}$ and $\sigma$ is the standard $p$th power Frobenius lift satisfying $\sigma^{a}=id$. Then the $a$th iteration of the Cartier operator $\mathscr{C}_{q}:=\mathscr{C}_{p}^{a}$ maps $\widehat{\Omega}_{f}$ to itself. It follows from the estimate (6) that modulo every power $p^{s}$ the image of $\mathscr{C}_{p}$ has finite rank. By this reason the trace of $\mathscr{C}_{q}$ is a well defined $p$-adic value. In this section we will prove the following

Theorem A.1**.**

The trace of $(q^{s}-1)^{n}\times\mathscr{C}_{q}^{s}$ on $\widehat{\Omega}_{f}$ equals the number of points on ${\mathbb{T}}^{n}\setminus Z_{f}$ with coordinates $x_{1},\ldots,x_{n}\in{\mathbb{F}}^{\times}_{q^{s}}$.

Remark A.2**.**

Since $\mathscr{C}_{q}$ is divisible by $q$ on the submodule of formal derivatives $U_{f}\subset\widehat{\Omega}_{f}$, we conclude from Theorem B.1 that on the quotient $Q_{f}=\widehat{\Omega}_{f}/U_{f}$ one has

[TABLE]

The term $1$ on the right corresponds to the eigenvector $1\in\Omega_{f}$, which has eigenvalue $1$. If the Hasse–Witt matrix $\beta_{p}(\Delta)$ is invertible, then $Q_{f}$ is a ${\mathbb{Z}}_{q}$-vector space of finite dimension and it follows from the above congruence that the polynomial

[TABLE]

raised to the power $(-1)^{n}$ is the unit-root part of the zeta function of the hypersuface $Z_{f}\subset{\mathbb{T}}^{n}$ over ${\mathbb{F}}_{q}$. Note that in our standard basis in $Q_{f}$ (that is, images of $\omega_{\bf u}\in\Omega_{f},{\bf u}\in\Delta_{\mathbb{Z}}$) the operator $\mathscr{C}_{q}$ is given by the transpose of

[TABLE]

where $\Lambda=\Lambda_{\sigma}$ is the matrix from Theorem 5.3.

We will use a resolution of the module $\Omega_{f}$. This construction ties our crystals with the exponential modules in the literature, e.g. in [1], and exhibits a natural lift of our Cartier operator which possesses nice properties and hence might be useful on its own. With the lifted Cartier operator, the point counting can be done using a version of Dwork’s trace formula, which is now a standard technique in $p$-adic analysis.

From now on $R$ is a characteristic zero ring, we will impose more assumptions when needed. Let us introduce the auxiliary variable $x_{0}$ and define the subring

[TABLE]

as the span of monomials ${\bf X}^{\bf u}=x_{0}^{u_{0}}\ldots x_{n}^{u_{n}}$ with ${\bf u}\in C(\Delta)_{\mathbb{Z}}$. We remind the reader that throughout the paper we denoted ${\bf x}^{{\bf u}}=x_{1}^{u_{1}}\ldots x_{n}^{u_{n}}$ for ${\bf u}=(u_{0},\ldots,u_{n})$, so now we shall use the capital letter for ${\bf X}^{\bf u}=x_{0}^{u_{0}}{\bf x}^{{\bf u}}$. We denote $F({\bf X})=x_{0}f({\bf x})\in R[\Delta]$. The operations

[TABLE]

are called twisted derivatives. Formally, we have $\mathscr{D}_{i,f}=e^{-F}\circ\theta_{i}\circ e^{F}$, where $\theta_{i}=x_{i}\frac{\partial}{\partial x_{i}}$. Twisted derivatives commute with each other.

Let $R[\Delta]^{+}$ be the free $R$-module generated by ${\bf X}^{\bf u}$ with ${\bf u}\in C(\Delta)_{Z}^{+}$. It is an ideal in $R[\Delta]$ and twisted derivatives preserve $R[\Delta]^{+}$. The Laplace transform is the $R$-linear map $\mathscr{R}:R[\Delta]^{+}\to\Omega_{f}$ given by

[TABLE]

This Laplace transform was basically defined in [7, p.244] and [1, §7]. See also our Remark 2.1. It is clear that $\mathscr{R}$ is surjective.

Proposition A.3**.**

The kernel of the Laplace transform is given by $\mathscr{D}_{0,f}(R[\Delta]^{+})$. Under the induced isomorphism

[TABLE]

the twisted defivative $\mathscr{D}_{i,f}$ corresponds to the usual derivative $\theta_{i}$ for each $1\leq i\leq n$.

Proof.

Under the Laplace transform the elements $\mathscr{D}_{0,f}({\bf X}^{\bf u})=u_{0}{\bf X}^{{\bf u}}+x_{0}f({\bf x}){\bf X}^{{\bf u}}$ are mapped to

[TABLE]

It is clear that these elements generate all relations in $\Omega_{f}$ and therefore they span the kernel of $\mathscr{R}$.

Let $1\leq i\leq n$. Since twisted derivatives commute, $\mathscr{D}_{i,f}$ maps ${\rm Ker}(\mathscr{R})={\rm Im}(\mathscr{D}_{0,f})$ to itself. The fact that the induced map on $\Omega_{f}$ coincides with $\theta_{i}$ can be easily checked on monomials:

[TABLE]

∎

Corollary A.4**.**

$R[\Delta]^{+}/\sum_{i=0}^{n}\mathscr{D}_{i,f}(R[\Delta]^{+})\cong\mathscr{W}_{f}$.

We would like to remark that in [1, Theorem 7.13] the quotient module on the left in this corollary was identified with $H^{n}_{dR}({\mathbb{T}}^{n}\setminus Z_{f})$ under the condition that $R$ is a field and $f({\bf x})$ is $\Delta$-regular. At the end of the introductory section we mentioned the relation between Dwork modules and de Rham cohomology having in mind Corollary B.4.

To define the Cartier operator, we turn on our usual assumptions that $\cap_{s}p^{s}R=\{0\}$ and $R$ is $p$-adically complete. Let $R\llbracket\Delta\rrbracket$ be the ring of formal power series with coefficients in $R$ and support in $C(\Delta)$. The $p$-adic completion $\widehat{R[\Delta]}=\underset{s}{\lim}R[\Delta]/p^{s}R[\Delta]$ consists of power series with infinitely growing $p$-adic valuation of coefficients:

[TABLE]

We denote by $\widehat{R[\Delta]}^{+}$ the ideal of power series with zero constant term ($a_{\bf 0}=0$). It follows from Proposition B.3 that \widehat{\Omega}_{f}\cong\widehat{R[\Delta]}^{+}/\mathscr{D}_{0,f}\Bigl{(}\widehat{R[\Delta]}^{+}\Bigr{)}.

Theorem A.5**.**

Consider the operator on power series given by

[TABLE]

Let $p>2$. For every $p$th power Frobenius lift $\sigma:R\to R$, the operator

[TABLE]

maps $R\llbracket\Delta\rrbracket$ to itself. Operator $\mathscr{V}_{\sigma}$ preserves $R\llbracket\Delta\rrbracket^{+}$ and it is divisible by $p$ on this submodule. The following commutation relation with twisted derivatives

[TABLE]

holds for each $0\leq i\leq n$.

The operator $\mathscr{V}_{\sigma}$ preserves $\widehat{R[\Delta]}$ and the induced map $p^{-1}\mathscr{V}_{\sigma}:\widehat{\Omega}_{f}\to\widehat{\Omega}_{f^{\sigma}}$ coincides with the Cartier operator $\mathscr{C}_{p}:\widehat{\Omega}_{f}\to\widehat{\Omega}_{f^{\sigma}}$ constructed in Section 3.

Proof.

Fix any ${\bf u}\in C(\Delta)_{\mathbb{Z}}^{+}$ and denote $\nu_{0}=p\lceil\frac{u_{0}}{p}\rceil-u_{0}$. Observe that

[TABLE]

and therefore

[TABLE]

where we substituted $f^{\sigma}({\bf x}^{p})=f({\bf x})^{p}+pG({\bf x})$ and recognised the sums

[TABLE]

as coefficients of the Dwork exponential $e^{x_{0}+\frac{x_{0}^{p}}{p}}=\sum_{n\geq 0}\gamma_{n}x_{0}^{n}$. The following standard estimate of their $p$-adic order

[TABLE]

implies that the matrix coefficients given by $\mathscr{V}_{\sigma}{\bf X}^{\bf u}=\sum_{{\bf v}}\mathscr{F}_{{\bf u},{\bf v}}{\bf X}^{\bf v}$ have $p$-adic valuations bounded by

[TABLE]

where we used $r=v_{0}-\lceil\frac{u_{0}}{p}\rceil$. Since this valuation is non-negative, we conclude that all $\mathscr{F}_{{\bf u},{\bf v}}\in R$ and hence $\mathscr{V}_{\sigma}$ maps the module of formal series $R\llbracket\Delta\rrbracket$ to itself. We also observe that $\mathscr{V}_{\sigma}$ is divisible by $p$ on $R\llbracket\Delta\rrbracket^{+}$. Moreover, $\mathscr{V}_{\sigma}$ preserves $\widehat{R[\Delta]}$ because when $v_{0}\to\infty$ we have $\mathrm{ord}_{p}(\mathscr{F}_{{\bf u},{\bf v}}\in R)\to\infty$ uniformly in $u_{0}$.

The commutation relation (35) follows immediately from $\theta_{i}\circ V_{p}=p\,\theta_{i}\circ V_{p}$ and the fact that $\mathscr{D}_{i,f}=e^{-F}\circ\theta_{i}\circ e^{F}$. Since $\widehat{\Omega}_{f}=\widehat{R[\Delta]}^{+}/\mathscr{D}_{0,f}(\widehat{R[\Delta]}^{+})$, we have a well defined induced map $p^{-1}\mathscr{V}_{\sigma}:\widehat{\Omega}_{f}\to\widehat{\Omega}_{f}^{\sigma}$. Let us show that this induced map coincides with the Cartier operator $\mathscr{C}_{p}$ defined in Proposition 3.3. Since $\mathscr{R}({\bf X}^{\bf w})=(-1)^{u_{0}}\omega_{{\bf w}}$, we consider

[TABLE]

where we used formula (5) and substitution $\frac{(u_{0}-1)!}{(\lceil\frac{u_{0}}{p}\rceil-1)!}=(-1)^{u_{0}}p^{\lceil\frac{u_{0}}{p}\rceil-1}\Gamma_{p}(u_{0})$. It is easy to check that the difference (37) equals

[TABLE]

where the coefficients $\mu_{s}\in{\mathbb{Q}}$ are determined by the recurrence

[TABLE]

(The initial term $\mu_{0}$ is also determined by this formula and convention $\mu_{-1}=0$.) We claim that

[TABLE]

and hence inside of $\mathscr{D}_{0,f^{\sigma}}(\ldots)$ in (38) the $p$-adic valuation of coefficients of the polynomial next to $x_{0}^{r+\lceil\frac{u_{0}}{p}\rceil}$ can be estimated from below as

[TABLE]

Since this valuation is non-negative and grows infinitely as $r\to\infty$, we conclude that (38) belongs to \mathscr{D}_{0,f^{\sigma}}\bigl{(}\widehat{R[\Delta]^{+}}\bigr{)} and therefore $p^{-1}\mathscr{V}_{\sigma}=\mathscr{C}_{p}$ on $\widehat{\Omega}_{f}$.

It only remains to prove (39). For this purpose we consider $f({\bf x})=1$ in (37) and (38). In this case $\mathscr{V}_{\sigma}$ is multiplication by the Dwork exponential $e^{x_{0}+p^{-1}x_{0}^{p}}$, followed by $\mathscr{C}_{p}$ and the substitution $x_{0}\mapsto-px_{0}$. We shall denote $\mathscr{V}_{\sigma}$ simply by $\mathscr{V}$ and $\mathscr{D}_{0,f^{\sigma}}=x_{0}\frac{d}{dx_{0}}+x_{0}$ by $\mathscr{D}$. The equality of (38) and (37) can be written as

[TABLE]

Note that $\mathscr{D}$ is invertible on ${\mathbb{Q}}_{p}\llbracket x_{0}\rrbracket^{+}=x_{0}{\mathbb{Q}}_{p}\llbracket x_{0}\rrbracket$, and hence the commutation relation $\mathscr{V}\mathscr{D}=p\mathscr{D}\mathscr{V}$ can be rewritten as $p^{-1}\mathscr{D}^{-1}\mathscr{V}=\mathscr{V}\mathscr{D}^{-1}$. Applying $\mathscr{D}^{-1}$ to the last identity and using the commutation relation, we get

[TABLE]

Let

[TABLE]

Let $\alpha_{0}=\frac{p-1}{p^{2}}-\frac{1}{p-1}$ and $\alpha_{1}=\frac{p-1}{p}-\frac{1}{p-1}$. Note that (39) precisely means that the series in (40) belongs to $L_{\alpha_{1},0}$. In order to demonstrate this fact, we first notice that for any $\alpha\geq\alpha_{0}$ and any $\beta$ we have $\mathscr{V}:L_{\alpha,\beta}\to L_{\alpha_{1},\beta}$. Indeed, since $e^{x_{0}+p^{-1}x_{0}^{p}}\in L_{\alpha_{0},0}$ we decompose $\mathscr{V}$ into three steps and check that

[TABLE]

In the view of (40), it now suffices to show that

[TABLE]

It is useful to observe that $\mathscr{D}(x_{0}^{m})=mx_{0}^{m}+x_{0}^{m+1}$ and

[TABLE]

Using these two rules one can easily check that

[TABLE]

and

[TABLE]

Note that under $\mathscr{D}(\ldots)$ the polynomial in (42) has integral coefficients and the series in (43) belongs to $L_{\alpha_{0},0}$ if one cuts off its constant term. We shall use (43) with $n=\lceil\frac{u_{0}}{p}\rceil$. Since $(-1)^{u_{0}}(u_{0}-1)!=\Gamma_{p}(u_{0})(\lceil\frac{u_{0}}{p}\rceil-1)!p^{\lceil\frac{u_{0}}{p}\rceil-1}$, we get

[TABLE]

Note that the constant term of the series in the right-hand side vanishes, which means that we integrated (41) in $x_{0}{\mathbb{Q}}_{p}\llbracket x_{0}\rrbracket$ explicitly. Since $\Gamma_{p}(u_{0})$ is a $p$-adic integer and $\alpha_{0}<0$, this series belongs to $L_{\alpha_{0},0}$ due to the remarks made after (42) and (43). This completes our proof of (39). ∎

Remark A.6**.**

One can easily define a connection on $R[\Delta]$ in a way and it commutes with the twisted derivatives. Namely, for every derivation $\delta:R\to R$ we define its action on $R[\Delta]$ as

[TABLE]

where the first summand simply means that the derivation $\delta$ is applied to the coefficients and the second one means multiplication by the polynomial $\delta F({\bf X})=x_{0}(\delta f)({\bf x})$. Formally, one can write $\nabla_{\delta}=e^{-F}\cdot\delta\cdot e^{F}$. To see that $\nabla_{\delta}$ commutes with the twisted derivatives, recall that $\mathscr{D}_{i,f}=e^{-F}\cdot\theta_{i}\cdot e^{F}$ and note that $\delta$ and $\theta_{i}$ commute. Operations $\nabla_{\delta}$ preserve $R[\Delta]^{+}$ and descend to its quotients by the images of twisted derivatives, particularly to $\Omega_{f}$ and $\mathscr{W}_{f}$. It is easy to check that $\nabla_{\delta}$ acts on $\Omega_{f}$ as the natural extension of $\delta$ to rational functions, the operation which we simply denoted by the same letter $\delta$ earlier in this paper.

Finally, observe that the operator $\mathscr{V}_{\sigma}=e^{-F^{\sigma}}\cdot V_{p}\cdot e^{F}$ defined in Theorem B.5 commutes with the connection operators. Namely, it is obvious that $V_{p}$ commutes with $\delta$ as operators on power series, and after twisting by exponentials we obtain

[TABLE]

where $\nabla^{\sigma}_{\delta}=\delta+\delta F^{\sigma}=e^{-F^{\sigma}}\cdot\delta\cdot e^{F^{\sigma}}$. This observation turns quotients of $\widehat{R[\Delta]}^{+}$ by twisted derivatives into crystals.

From now on we consider $R={\mathbb{Z}}_{q}$ with $q=p^{a}$. Here we have the standard $p$th power Frobenius lift $\sigma:{\mathbb{Z}}_{q}\to{\mathbb{Z}}_{q}$ which satisfies $\sigma^{a}=id$. Consider the operator on power series $Z_{q}\llbracket\Delta\rrbracket$ given by

[TABLE]

Below we compute the traces of powers of $\mathscr{V}_{q}$ using a few standard tricks in $p$-adic analysis, which are basically due to Dwork.

Remark A.7**.**

The traces are well-defined $p$-adic numbers because modulo every $p^{s}$ the operator $\mathscr{V}_{q}$ has finite-dimensional image (see the $p$-adic estimate of the matrix entries in the proof of Theorem B.5). Note also that the traces only depend on the mod $p$ reduction of the polynomial $F({\bf X})=x_{0}f({\bf x})$. Indeed, if $F^{\prime}({\bf X})-F({\bf X})=pG({\bf X})$ then the respective operators on power series are conjugate $\mathscr{V}_{q}^{\prime}=e^{-pG}\circ\mathscr{V}_{q}\circ e^{pG}$ and modulo each power of $p$ this identity can be written using matrices of finite size.

Proposition A.8**.**

For all $s\geq 1$ one has

[TABLE]

Proof.

Let $\pi\in\overline{{\mathbb{Q}}_{p}}$ be a number satisfying $\pi^{p-1}=-p$. We will work with Laurent series with coefficients in $R={\mathbb{Q}}_{q}(\pi)$ and support in the cone $C(\Delta)$. Let $\rho:R\llbracket\Delta\rrbracket\to R\llbracket\Delta\rrbracket$ be the operation given by $\rho({\bf X}^{\bf u})=\pi^{u_{0}}{\bf X}^{\bf u}$. Note that $V_{p}=\rho^{-1}\circ\mathscr{C}_{p}\circ\rho$ and $e^{\pm F}=\rho^{-1}\circ e^{\pm\pi F}\circ\rho$, and hence

[TABLE]

where we used the power series

[TABLE]

From (44) it is clear that ${\rm Tr}(\mathscr{V}_{q}^{s})=\sum_{{\bf u}\in C(\Delta)_{\mathbb{Z}}}b_{(q^{s}-1){\bf u}}$. Since

[TABLE]

this trace can be computed by summation of values of (45) over tuples of Teichmüller units in ${\mathbb{Z}}_{q^{s}}$:

[TABLE]

To evaluate the sum on the left, consider the Dwork exponential $\theta_{p}(z)=\exp(\pi z-\pi z^{p})$. This series has $p$-adic radius of convergence $>1$ and $\zeta_{p}:=\theta_{p}(1)=1+\pi({\rm mod}\ \pi^{2})$ is a $p$th root of unity. For $k\geq 1$, let $\theta_{p^{k}}(z)=\exp(\pi z-\pi z^{p^{k}})=\prod_{i=1}^{k-1}\theta_{p}(z^{p^{i}})$. The additive character $\psi_{k}:{\mathbb{F}}_{p^{k}}\to{\mathbb{Q}}_{p}(\pi)^{\times}$ given by $\psi_{k}(\bar{x})=\zeta_{p}^{{\rm Tr}_{{\mathbb{F}}_{p^{k}}/{\mathbb{F}}_{p}}(\bar{x})}$ is related to the Dwork exponential via $\psi_{k}(\bar{x})=\theta_{p^{k}}({\rm Teich}(\bar{x}))$.

Write $F({\bf X})=x_{0}f({\bf x})=\sum a_{\bf u}{\bf X}^{\bf u}$ and let $\bar{F}({\bf X})=\sum\bar{a}_{\bf u}{\bf X}^{\bf u}$ with $\bar{a}_{\bf u}\in{\mathbb{F}}_{q}$ be the reduction of $F$ modulo $p$. Denote $a_{{\bf u}}^{\prime}={\rm Teich}(\bar{a}_{\bf u})$ and $F^{\prime}({\bf X})=\sum a_{\bf u}^{\prime}{\bf X}^{\bf u}$. For any vector $\bar{X}=(\bar{x}_{0},\ldots,\bar{x}_{n})\in{\mathbb{F}}_{q^{s}}^{n+1}$ we have

[TABLE]

Therefore the left-hand sum in (46) for $F^{\prime}$ can be evaluated as

[TABLE]

By Remark B.7, since $F^{\prime}\equiv F({\rm mod}\ p)$ traces of powers of $\mathscr{V}_{q}^{\prime}=e^{-F^{\prime}}\circ V_{p}^{a}\circ e^{F^{\prime}}$ and $\mathscr{V}_{q}=e^{-F}\circ V_{p}^{a}\circ e^{F}$ are equal. Hence our claim follows from (46) and (47). ∎

Proof of Theorem B.1..

By Theorem B.5, we have

[TABLE]

Here the second equality follows from the commutation relation $\mathscr{V}_{q}^{s}\circ\mathscr{D}_{0,f}=q^{s}\,\mathscr{D}_{0,f}\circ\mathscr{V}_{q}^{s}$. Traces on $\widehat{R[\Delta]}^{+}$ and $R\llbracket\Delta\rrbracket^{+}$ are the same. It is clear from the definition of $\mathscr{V}_{q}$ that for every $s\geq 1$ one has $\mathscr{V}_{q}^{s}({\bf X}^{\bf 0})={\bf X}^{\bf 0}+$ terms with $u_{0}\geq 1$, and hence ${\rm Tr}(\mathscr{V}_{q}^{s}|R\llbracket\Delta\rrbracket^{+})\;=\;{\rm Tr}(\mathscr{V}_{q}^{s}|R\llbracket\Delta\rrbracket)-1$. Finally, we combine (48) with Proposition B.8 and get

[TABLE]

∎

Appendix B Point counting and an alternative construction of the Cartier operator

Suppose that $R={\mathbb{Z}}_{q}$ where $q=p^{a}$ and $\sigma$ is the standard $p$th power Frobenius lift satisfying $\sigma^{a}=id$. Then the $a$th iteration of the Cartier operator $\mathscr{C}_{q}:=\mathscr{C}_{p}^{a}$ maps $\widehat{\Omega}_{f}$ to itself. It follows from the estimate (6) that modulo every power $p^{s}$ the image of $\mathscr{C}_{p}$ has finite rank. By this reason the trace of $\mathscr{C}_{q}$ is a well defined $p$-adic value. In this section we will prove the following

Theorem B.1**.**

The trace of $(q^{s}-1)^{n}\times\mathscr{C}_{q}^{s}$ on $\widehat{\Omega}_{f}$ equals the number of points on ${\mathbb{T}}^{n}\setminus Z_{f}$ with coordinates $x_{1},\ldots,x_{n}\in{\mathbb{F}}^{\times}_{q^{s}}$.

Remark B.2**.**

Since $\mathscr{C}_{q}$ is divisible by $q$ on the submodule of formal derivatives $U_{f}\subset\widehat{\Omega}_{f}$, we conclude from Theorem B.1 that on the quotient $Q_{f}=\widehat{\Omega}_{f}/U_{f}$ one has

[TABLE]

The term $1$ on the right corresponds to the eigenvector $1\in\Omega_{f}$, which has eigenvalue $1$. If the Hasse–Witt matrix $\beta_{p}(\Delta)$ is invertible, then $Q_{f}$ is a ${\mathbb{Z}}_{q}$-vector space of finite dimension and it follows from the above congruence that the polynomial

[TABLE]

raised to the power $(-1)^{n}$ is the unit-root part of the zeta function of the hypersuface $Z_{f}\subset{\mathbb{T}}^{n}$ over ${\mathbb{F}}_{q}$. Note that in our standard basis in $Q_{f}$ (that is, images of $\omega_{\bf u}\in\Omega_{f},{\bf u}\in\Delta_{\mathbb{Z}}$) the operator $\mathscr{C}_{q}$ is given by the transpose of

[TABLE]

where $\Lambda=\Lambda_{\sigma}$ is the matrix from Theorem 5.3.

We will use a resolution of the module $\Omega_{f}$. This construction ties our crystals with the exponential modules in the literature, e.g. in [1], and exhibits a natural lift of our Cartier operator which possesses nice properties and hence might be useful on its own. With the lifted Cartier operator, the point counting can be done using a version of Dwork’s trace formula, which is now a standard technique in $p$-adic analysis.

From now on $R$ is a characteristic zero ring, we will impose more assumptions when needed. Let us introduce the auxiliary variable $x_{0}$ and define the subring

[TABLE]

as the span of monomials ${\bf X}^{\bf u}=x_{0}^{u_{0}}\ldots x_{n}^{u_{n}}$ with ${\bf u}\in C(\Delta)_{\mathbb{Z}}$. We remind the reader that throughout the paper we denoted ${\bf x}^{{\bf u}}=x_{1}^{u_{1}}\ldots x_{n}^{u_{n}}$ for ${\bf u}=(u_{0},\ldots,u_{n})$, so now we shall use the capital letter for ${\bf X}^{\bf u}=x_{0}^{u_{0}}{\bf x}^{{\bf u}}$. We denote $F({\bf X})=x_{0}f({\bf x})\in R[\Delta]$. The operations

[TABLE]

are called twisted derivatives. Formally, we have $\mathscr{D}_{i,f}=e^{-F}\circ\theta_{i}\circ e^{F}$, where $\theta_{i}=x_{i}\frac{\partial}{\partial x_{i}}$. Twisted derivatives commute with each other.

Let $R[\Delta]^{+}$ be the free $R$-module generated by ${\bf X}^{\bf u}$ with ${\bf u}\in C(\Delta)_{Z}^{+}$. It is an ideal in $R[\Delta]$ and twisted derivatives preserve $R[\Delta]^{+}$. The Laplace transform is the $R$-linear map $\mathscr{R}:R[\Delta]^{+}\to\Omega_{f}$ given by

[TABLE]

This Laplace transform was basically defined in [7, p.244] and [1, §7]. See also our Remark 2.1. It is clear that $\mathscr{R}$ is surjective.

Proposition B.3**.**

The kernel of the Laplace transform is given by $\mathscr{D}_{0,f}(R[\Delta]^{+})$. Under the induced isomorphism

[TABLE]

the twisted defivative $\mathscr{D}_{i,f}$ corresponds to the usual derivative $\theta_{i}$ for each $1\leq i\leq n$.

Proof.

Under the Laplace transform the elements $\mathscr{D}_{0,f}({\bf X}^{\bf u})=u_{0}{\bf X}^{{\bf u}}+x_{0}f({\bf x}){\bf X}^{{\bf u}}$ are mapped to

[TABLE]

It is clear that these elements generate all relations in $\Omega_{f}$ and therefore they span the kernel of $\mathscr{R}$.

Let $1\leq i\leq n$. Since twisted derivatives commute, $\mathscr{D}_{i,f}$ maps ${\rm Ker}(\mathscr{R})={\rm Im}(\mathscr{D}_{0,f})$ to itself. The fact that the induced map on $\Omega_{f}$ coincides with $\theta_{i}$ can be easily checked on monomials:

[TABLE]

∎

Corollary B.4**.**

$R[\Delta]^{+}/\sum_{i=0}^{n}\mathscr{D}_{i,f}(R[\Delta]^{+})\cong\mathscr{W}_{f}$.

We would like to remark that in [1, Theorem 7.13] the quotient module on the left in this corollary was identified with $H^{n}_{dR}({\mathbb{T}}^{n}\setminus Z_{f})$ under the condition that $R$ is a field and $f({\bf x})$ is $\Delta$-regular. At the end of the introductory section we mentioned the relation between Dwork modules and de Rham cohomology having in mind Corollary B.4.

To define the Cartier operator, we turn on our usual assumptions that $\cap_{s}p^{s}R=\{0\}$ and $R$ is $p$-adically complete. Let $R\llbracket\Delta\rrbracket$ be the ring of formal power series with coefficients in $R$ and support in $C(\Delta)$. The $p$-adic completion $\widehat{R[\Delta]}=\underset{s}{\lim}R[\Delta]/p^{s}R[\Delta]$ consists of power series with infinitely growing $p$-adic valuation of coefficients:

[TABLE]

We denote by $\widehat{R[\Delta]}^{+}$ the ideal of power series with zero constant term ($a_{\bf 0}=0$). It follows from Proposition B.3 that \widehat{\Omega}_{f}\cong\widehat{R[\Delta]}^{+}/\mathscr{D}_{0,f}\Bigl{(}\widehat{R[\Delta]}^{+}\Bigr{)}.

Theorem B.5**.**

Consider the operator on power series given by

[TABLE]

Let $p>2$. For every $p$th power Frobenius lift $\sigma:R\to R$, the operator

[TABLE]

maps $R\llbracket\Delta\rrbracket$ to itself. Operator $\mathscr{V}_{\sigma}$ preserves $R\llbracket\Delta\rrbracket^{+}$ and it is divisible by $p$ on this submodule. The following commutation relation with twisted derivatives

[TABLE]

holds for each $0\leq i\leq n$.

The operator $\mathscr{V}_{\sigma}$ preserves $\widehat{R[\Delta]}$ and the induced map $p^{-1}\mathscr{V}_{\sigma}:\widehat{\Omega}_{f}\to\widehat{\Omega}_{f^{\sigma}}$ coincides with the Cartier operator $\mathscr{C}_{p}:\widehat{\Omega}_{f}\to\widehat{\Omega}_{f^{\sigma}}$ constructed in Section 3.

Proof.

Fix any ${\bf u}\in C(\Delta)_{\mathbb{Z}}^{+}$ and denote $\nu_{0}=p\lceil\frac{u_{0}}{p}\rceil-u_{0}$. Observe that

[TABLE]

and therefore

[TABLE]

where we substituted $f^{\sigma}({\bf x}^{p})=f({\bf x})^{p}+pG({\bf x})$ and recognised the sums

[TABLE]

as coefficients of the Dwork exponential $e^{x_{0}+\frac{x_{0}^{p}}{p}}=\sum_{n\geq 0}\gamma_{n}x_{0}^{n}$. The following standard estimate of their $p$-adic order

[TABLE]

implies that the matrix coefficients given by $\mathscr{V}_{\sigma}{\bf X}^{\bf u}=\sum_{{\bf v}}\mathscr{F}_{{\bf u},{\bf v}}{\bf X}^{\bf v}$ have $p$-adic valuations bounded by

[TABLE]

where we used $r=v_{0}-\lceil\frac{u_{0}}{p}\rceil$. Since this valuation is non-negative, we conclude that all $\mathscr{F}_{{\bf u},{\bf v}}\in R$ and hence $\mathscr{V}_{\sigma}$ maps the module of formal series $R\llbracket\Delta\rrbracket$ to itself. We also observe that $\mathscr{V}_{\sigma}$ is divisible by $p$ on $R\llbracket\Delta\rrbracket^{+}$. Moreover, $\mathscr{V}_{\sigma}$ preserves $\widehat{R[\Delta]}$ because when $v_{0}\to\infty$ we have $\mathrm{ord}_{p}(\mathscr{F}_{{\bf u},{\bf v}}\in R)\to\infty$ uniformly in $u_{0}$.

The commutation relation (35) follows immediately from $\theta_{i}\circ V_{p}=p\,\theta_{i}\circ V_{p}$ and the fact that $\mathscr{D}_{i,f}=e^{-F}\circ\theta_{i}\circ e^{F}$. Since $\widehat{\Omega}_{f}=\widehat{R[\Delta]}^{+}/\mathscr{D}_{0,f}(\widehat{R[\Delta]}^{+})$, we have a well defined induced map $p^{-1}\mathscr{V}_{\sigma}:\widehat{\Omega}_{f}\to\widehat{\Omega}_{f}^{\sigma}$. Let us show that this induced map coincides with the Cartier operator $\mathscr{C}_{p}$ defined in Proposition 3.3. Since $\mathscr{R}({\bf X}^{\bf w})=(-1)^{u_{0}}\omega_{{\bf w}}$, we consider

[TABLE]

where we used formula (5) and substitution $\frac{(u_{0}-1)!}{(\lceil\frac{u_{0}}{p}\rceil-1)!}=(-1)^{u_{0}}p^{\lceil\frac{u_{0}}{p}\rceil-1}\Gamma_{p}(u_{0})$. It is easy to check that the difference (37) equals

[TABLE]

where the coefficients $\mu_{s}\in{\mathbb{Q}}$ are determined by the recurrence

[TABLE]

(The initial term $\mu_{0}$ is also determined by this formula and convention $\mu_{-1}=0$.) We claim that

[TABLE]

and hence inside of $\mathscr{D}_{0,f^{\sigma}}(\ldots)$ in (38) the $p$-adic valuation of coefficients of the polynomial next to $x_{0}^{r+\lceil\frac{u_{0}}{p}\rceil}$ can be estimated from below as

[TABLE]

Since this valuation is non-negative and grows infinitely as $r\to\infty$, we conclude that (38) belongs to \mathscr{D}_{0,f^{\sigma}}\bigl{(}\widehat{R[\Delta]^{+}}\bigr{)} and therefore $p^{-1}\mathscr{V}_{\sigma}=\mathscr{C}_{p}$ on $\widehat{\Omega}_{f}$.

It only remains to prove (39). For this purpose we consider $f({\bf x})=1$ in (37) and (38). In this case $\mathscr{V}_{\sigma}$ is multiplication by the Dwork exponential $e^{x_{0}+p^{-1}x_{0}^{p}}$, followed by $\mathscr{C}_{p}$ and the substitution $x_{0}\mapsto-px_{0}$. We shall denote $\mathscr{V}_{\sigma}$ simply by $\mathscr{V}$ and $\mathscr{D}_{0,f^{\sigma}}=x_{0}\frac{d}{dx_{0}}+x_{0}$ by $\mathscr{D}$. The equality of (38) and (37) can be written as

[TABLE]

Note that $\mathscr{D}$ is invertible on ${\mathbb{Q}}_{p}\llbracket x_{0}\rrbracket^{+}=x_{0}{\mathbb{Q}}_{p}\llbracket x_{0}\rrbracket$, and hence the commutation relation $\mathscr{V}\mathscr{D}=p\mathscr{D}\mathscr{V}$ can be rewritten as $p^{-1}\mathscr{D}^{-1}\mathscr{V}=\mathscr{V}\mathscr{D}^{-1}$. Applying $\mathscr{D}^{-1}$ to the last identity and using the commutation relation, we get

[TABLE]

Let

[TABLE]

Let $\alpha_{0}=\frac{p-1}{p^{2}}-\frac{1}{p-1}$ and $\alpha_{1}=\frac{p-1}{p}-\frac{1}{p-1}$. Note that (39) precisely means that the series in (40) belongs to $L_{\alpha_{1},0}$. In order to demonstrate this fact, we first notice that for any $\alpha\geq\alpha_{0}$ and any $\beta$ we have $\mathscr{V}:L_{\alpha,\beta}\to L_{\alpha_{1},\beta}$. Indeed, since $e^{x_{0}+p^{-1}x_{0}^{p}}\in L_{\alpha_{0},0}$ we decompose $\mathscr{V}$ into three steps and check that

[TABLE]

In the view of (40), it now suffices to show that

[TABLE]

It is useful to observe that $\mathscr{D}(x_{0}^{m})=mx_{0}^{m}+x_{0}^{m+1}$ and

[TABLE]

Using these two rules one can easily check that

[TABLE]

and

[TABLE]

Note that under $\mathscr{D}(\ldots)$ the polynomial in (42) has integral coefficients and the series in (43) belongs to $L_{\alpha_{0},0}$ if one cuts off its constant term. We shall use (43) with $n=\lceil\frac{u_{0}}{p}\rceil$. Since $(-1)^{u_{0}}(u_{0}-1)!=\Gamma_{p}(u_{0})(\lceil\frac{u_{0}}{p}\rceil-1)!p^{\lceil\frac{u_{0}}{p}\rceil-1}$, we get

[TABLE]

Note that the constant term of the series in the right-hand side vanishes, which means that we integrated (41) in $x_{0}{\mathbb{Q}}_{p}\llbracket x_{0}\rrbracket$ explicitly. Since $\Gamma_{p}(u_{0})$ is a $p$-adic integer and $\alpha_{0}<0$, this series belongs to $L_{\alpha_{0},0}$ due to the remarks made after (42) and (43). This completes our proof of (39). ∎

Remark B.6**.**

One can easily define a connection on $R[\Delta]$ in a way and it commutes with the twisted derivatives. Namely, for every derivation $\delta:R\to R$ we define its action on $R[\Delta]$ as

[TABLE]

where the first summand simply means that the derivation $\delta$ is applied to the coefficients and the second one means multiplication by the polynomial $\delta F({\bf X})=x_{0}(\delta f)({\bf x})$. Formally, one can write $\nabla_{\delta}=e^{-F}\cdot\delta\cdot e^{F}$. To see that $\nabla_{\delta}$ commutes with the twisted derivatives, recall that $\mathscr{D}_{i,f}=e^{-F}\cdot\theta_{i}\cdot e^{F}$ and note that $\delta$ and $\theta_{i}$ commute. Operations $\nabla_{\delta}$ preserve $R[\Delta]^{+}$ and descend to its quotients by the images of twisted derivatives, particularly to $\Omega_{f}$ and $\mathscr{W}_{f}$. It is easy to check that $\nabla_{\delta}$ acts on $\Omega_{f}$ as the natural extension of $\delta$ to rational functions, the operation which we simply denoted by the same letter $\delta$ earlier in this paper.

Finally, observe that the operator $\mathscr{V}_{\sigma}=e^{-F^{\sigma}}\cdot V_{p}\cdot e^{F}$ defined in Theorem B.5 commutes with the connection operators. Namely, it is obvious that $V_{p}$ commutes with $\delta$ as operators on power series, and after twisting by exponentials we obtain

[TABLE]

where $\nabla^{\sigma}_{\delta}=\delta+\delta F^{\sigma}=e^{-F^{\sigma}}\cdot\delta\cdot e^{F^{\sigma}}$. This observation turns quotients of $\widehat{R[\Delta]}^{+}$ by twisted derivatives into crystals.

From now on we consider $R={\mathbb{Z}}_{q}$ with $q=p^{a}$. Here we have the standard $p$th power Frobenius lift $\sigma:{\mathbb{Z}}_{q}\to{\mathbb{Z}}_{q}$ which satisfies $\sigma^{a}=id$. Consider the operator on power series $Z_{q}\llbracket\Delta\rrbracket$ given by

[TABLE]

Below we compute the traces of powers of $\mathscr{V}_{q}$ using a few standard tricks in $p$-adic analysis, which are basically due to Dwork.

Remark B.7**.**

The traces are well-defined $p$-adic numbers because modulo every $p^{s}$ the operator $\mathscr{V}_{q}$ has finite-dimensional image (see the $p$-adic estimate of the matrix entries in the proof of Theorem B.5). Note also that the traces only depend on the mod $p$ reduction of the polynomial $F({\bf X})=x_{0}f({\bf x})$. Indeed, if $F^{\prime}({\bf X})-F({\bf X})=pG({\bf X})$ then the respective operators on power series are conjugate $\mathscr{V}_{q}^{\prime}=e^{-pG}\circ\mathscr{V}_{q}\circ e^{pG}$ and modulo each power of $p$ this identity can be written using matrices of finite size.

Proposition B.8**.**

For all $s\geq 1$ one has

[TABLE]

Proof.

Let $\pi\in\overline{{\mathbb{Q}}_{p}}$ be a number satisfying $\pi^{p-1}=-p$. We will work with Laurent series with coefficients in $R={\mathbb{Q}}_{q}(\pi)$ and support in the cone $C(\Delta)$. Let $\rho:R\llbracket\Delta\rrbracket\to R\llbracket\Delta\rrbracket$ be the operation given by $\rho({\bf X}^{\bf u})=\pi^{u_{0}}{\bf X}^{\bf u}$. Note that $V_{p}=\rho^{-1}\circ\mathscr{C}_{p}\circ\rho$ and $e^{\pm F}=\rho^{-1}\circ e^{\pm\pi F}\circ\rho$, and hence

[TABLE]

where we used the power series

[TABLE]

From (44) it is clear that ${\rm Tr}(\mathscr{V}_{q}^{s})=\sum_{{\bf u}\in C(\Delta)_{\mathbb{Z}}}b_{(q^{s}-1){\bf u}}$. Since

[TABLE]

this trace can be computed by summation of values of (45) over tuples of Teichmüller units in ${\mathbb{Z}}_{q^{s}}$:

[TABLE]

To evaluate the sum on the left, consider the Dwork exponential $\theta_{p}(z)=\exp(\pi z-\pi z^{p})$. This series has $p$-adic radius of convergence $>1$ and $\zeta_{p}:=\theta_{p}(1)=1+\pi({\rm mod}\ \pi^{2})$ is a $p$th root of unity. For $k\geq 1$, let $\theta_{p^{k}}(z)=\exp(\pi z-\pi z^{p^{k}})=\prod_{i=1}^{k-1}\theta_{p}(z^{p^{i}})$. The additive character $\psi_{k}:{\mathbb{F}}_{p^{k}}\to{\mathbb{Q}}_{p}(\pi)^{\times}$ given by $\psi_{k}(\bar{x})=\zeta_{p}^{{\rm Tr}_{{\mathbb{F}}_{p^{k}}/{\mathbb{F}}_{p}}(\bar{x})}$ is related to the Dwork exponential via $\psi_{k}(\bar{x})=\theta_{p^{k}}({\rm Teich}(\bar{x}))$.

Write $F({\bf X})=x_{0}f({\bf x})=\sum a_{\bf u}{\bf X}^{\bf u}$ and let $\bar{F}({\bf X})=\sum\bar{a}_{\bf u}{\bf X}^{\bf u}$ with $\bar{a}_{\bf u}\in{\mathbb{F}}_{q}$ be the reduction of $F$ modulo $p$. Denote $a_{{\bf u}}^{\prime}={\rm Teich}(\bar{a}_{\bf u})$ and $F^{\prime}({\bf X})=\sum a_{\bf u}^{\prime}{\bf X}^{\bf u}$. For any vector $\bar{X}=(\bar{x}_{0},\ldots,\bar{x}_{n})\in{\mathbb{F}}_{q^{s}}^{n+1}$ we have

[TABLE]

Therefore the left-hand sum in (46) for $F^{\prime}$ can be evaluated as

[TABLE]

By Remark B.7, since $F^{\prime}\equiv F({\rm mod}\ p)$ traces of powers of $\mathscr{V}_{q}^{\prime}=e^{-F^{\prime}}\circ V_{p}^{a}\circ e^{F^{\prime}}$ and $\mathscr{V}_{q}=e^{-F}\circ V_{p}^{a}\circ e^{F}$ are equal. Hence our claim follows from (46) and (47). ∎

Proof of Theorem B.1..

By Theorem B.5, we have

[TABLE]

Here the second equality follows from the commutation relation $\mathscr{V}_{q}^{s}\circ\mathscr{D}_{0,f}=q^{s}\,\mathscr{D}_{0,f}\circ\mathscr{V}_{q}^{s}$. Traces on $\widehat{R[\Delta]}^{+}$ and $R\llbracket\Delta\rrbracket^{+}$ are the same. It is clear from the definition of $\mathscr{V}_{q}$ that for every $s\geq 1$ one has $\mathscr{V}_{q}^{s}({\bf X}^{\bf 0})={\bf X}^{\bf 0}+$ terms with $u_{0}\geq 1$, and hence ${\rm Tr}(\mathscr{V}_{q}^{s}|R\llbracket\Delta\rrbracket^{+})\;=\;{\rm Tr}(\mathscr{V}_{q}^{s}|R\llbracket\Delta\rrbracket)-1$. Finally, we combine (48) with Proposition B.8 and get

[TABLE]

∎