A generalization of the Hasse-Witt matrix of a hypersurface

TL;DR

This paper extends the classical Hasse-Witt matrix to provide meaningful mod p information about the zeta function of hypersurfaces of any degree, including cases where the traditional matrix is trivial.

Contribution

The authors generalize the classical Hasse-Witt matrix to yield nontrivial zeta function congruences for all degrees, and analyze its differential equations and invertibility.

Findings

The generalized matrix provides nontrivial mod p zeta function information for all degrees.

The matrix satisfies specific differential equations.

It is shown to be generically invertible.

Abstract

The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite field of characteristic $p$ gives essentially complete mod $p$ information about the zeta function of the hypersurface. But if the degree $d$ of the hypersurface is $\leq n$, the zeta function is trivial mod $p$ and the Hasse-Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse-Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all $d$. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.

Equations

$$f_{\lambda }(x)=\sum _{j=1}^N{\lambda }_j{x}^{{\mathbf{a}}_j}\in {\mathbb{F}}_q[x_0,\dots ,x_n]$$

$$Z(X_{\lambda }/{\mathbb{F}}_q,t)=\frac{P_{\lambda }(t{)}^{(-1{)}^n}}{(1-t)(1-qt)\cdots (1-q^{n-1}t)}$$

$$\bigg{\lceil}\frac{n+1}{d}\bigg{\rceil}=\mu+1,$$

$$P_{\lambda }(q^{-\mu }t)\in 1+t\mathbb{Z}[[t]].$$

$${\mathbf{a}}_j^{+}=(a_{0j},\dots ,a_{nj},1)\in {\mathbb{N}}^{n+2},j=1,\dots ,N,$$

$$U=\bigg{\{}u=(u_{0},\dots,u_{n+1})\in{\mathbb{N}}^{n+2}\mid\text{$\sum_{i=0}^{n}u_{i}=du_{n+1}$ and $u_{i}>0$ for all $i$}\bigg{\}}.$$

$$U_{\min }=\{u=(u_0,\dots ,u_{n+1})\in U\mid u_{n+1}=\mu +1\},$$

$$A_{uv}(\Lambda )=(-1{)}^{\mu +1}\sum _{\begin{array}{c}\nu \in {\mathbb{N}}^N\\ {\sum }_{j=1}^N{\nu }_j{\mathbf{a}}_j^{+}=pu-v\end{array}}\frac{{\Lambda }_1^{{\nu }_1}\cdots {\Lambda }_N^{{\nu }_N}}{{\nu }_1!\cdots {\nu }_N!}\in \mathbb{Q}[{\Lambda }_1,\dots ,{\Lambda }_N].$$

$$\sum _{j=1}^N{\nu }_j=(p-1)(\mu +1).$$

$$P_{\lambda }(t)=\frac{Q_{\lambda }(t)}{R_{\lambda }(t)},$$

$$Q_{\lambda }(q^{-\mu }t),R_{\lambda }(q^{-\mu }t)\in 1+t\mathbb{Z}[t].$$

$$Q_{\lambda}(q^{-\mu}t)\equiv\det\big{(}I-t\bar{A}(\lambda^{p^{a-1}})\bar{A}(\lambda^{p^{a-2}})\cdots\bar{A}(\lambda)\big{)}\pmod{p}.$$

$$\bigg{\lceil}\frac{|I|}{d}\bigg{\rceil}=\mu_{I}+1.$$

$$U^{I}=\bigg{\{}u=(u_{0},\dots,u_{n+1})\in{\mathbb{N}}^{n+2}\mid\text{$\sum_{i=0}^{n}u_{i}=du_{n+1}$ and $u_{i}>0$ for all $i\in I$}\bigg{\}}.$$

$$U_{\min }^I=\{u=(u_0,\dots ,u_{n+1})\in U^I\mid u_{n+1}={\mu }_I+1\},$$

$$a_{i_{0}k}\ge \{\begin{array}{ll}{u}_{i_0}& \text{if i0∈I,}\\ u_{i_0}+1& \text{if i0∈I.}\end{array}$$

$$u-{\mathbf{a}}_k^{+}=(u_0-a_{0k},\dots ,u_n-a_{nk},{\mu }_I)\in U^I,$$

$${\nu }_k{a}_{i_{0}k}\ge p{a}_{i_{0}k}\ge \{\begin{array}{ll}p{u}_{i_0}& \text{if i0∈I,}\\ p{u}_{i_0}+p& \text{if i0∈I,}\end{array}$$

$${\nu }_k{a}_{i_{0}k}>p{u}_{i_0}-v_{i_0}.$$

$${\nu }_k{a}_{i_{0}k}\le p{u}_{i_0}-v_{i_0}.$$

$$L=\bigg{\{}l=(l_{1},\dots,l_{N})\in{\mathbb{Z}}^{N}\mid\sum_{j=1}^{N}l_{j}{\bf a}_{j}^{+}={\bf 0}\bigg{\}},$$

$$\Box_{l}=\prod_{l_{j}>0}\bigg{(}\frac{\partial}{\partial\Lambda_{j}}\bigg{)}^{l_{j}}-\prod_{l_{j}<0}\bigg{(}\frac{\partial}{\partial\Lambda_{j}}\bigg{)}^{-l_{j}}\quad\text{for $l\in L$}$$

$$Z_i=\sum _{j=1}^N{a}_{ij}{\Lambda }_j\frac{\partial }{\partial {\Lambda }_j}-{\beta }_i\text{for i=0,…,n}$$

$$Z_{n+1}=\sum _{j=1}^N{\Lambda }_j\frac{\partial }{\partial {\Lambda }_j}-{\beta }_{n+1}.$$

$$A_{uv}^I(\Lambda )=(-1{)}^{{\mu }_I+1}\sum _{\begin{array}{c}\nu \in {\mathbb{N}}^N\\ {\sum }_{j=1}^N{\nu }_j{\mathbf{a}}_j^{+}=pu-v\end{array}}\frac{{\Lambda }_1^{{\nu }_1}\cdots {\Lambda }_N^{{\nu }_N}}{{\nu }_1!\cdots {\nu }_N!}\in \mathbb{Q}[{\Lambda }_1,\dots ,{\Lambda }_N].$$

$$L_{0}=\bigg{\{}\sum_{u\in{\mathbb{N}}^{n+2}}c_{u}\gamma_{0}^{pu_{n+1}}x^{u}\mid\text{$\sum_{i=0}^{n}u_{i}-du_{n+1}=0$, $c_{u}\in{\mathbb{C}}_{p}$, and $\{c_{u}\}$ is bounded}\bigg{\}}.$$

$$L_{0}^{I}=\bigg{\{}\sum_{u\in{\mathbb{N}}^{n+2}}c_{u}\gamma_{0}^{pu_{n+1}}x^{u}\in L_{0}\mid\text{$u_{i}>0$ for $i\in I$}\bigg{\}}.$$

$$\theta (t)=\mathrm{AH}({\gamma }_{0}t)=\sum _{i=0}^{\infty }{\theta }_i{t}^i.$$

$${\mathrm{ord}}_p{\theta }_i\ge \frac{i}{p-1}.$$

$$\theta (\hat{\lambda },x)=\prod _{j=1}^N\theta ({\hat{\lambda }}_j{x}^{{\mathbf{a}}_j^{+}}),$$

Full text

1. Introduction

Let $p$ be a prime number, let $q=p^{a}$, and let ${\mathbb{F}}_{q}$ be the field of $q$ elements. Let

[TABLE]

be a homogeneous polynomial of degree $d$. We write ${\bf a}_{j}=(a_{0j},\dots,a_{nj})$ with $\sum_{i=0}^{n}a_{ij}=d$. Let $X_{\lambda}\subseteq{\mathbb{P}}^{n}$ be the hypersurface defined by the equation $f_{\lambda}(x)=0$ and let $Z(X_{\lambda}/{\mathbb{F}}_{q},t)$ be its zeta function. We write

[TABLE]

for some rational function $P_{\lambda}(t)\in 1+t{\mathbb{Z}}[[t]]$. If $d=1$ then $P_{\lambda}(t)=1$, so we shall always assume that $d\geq 2$.

Define a nonnegative integer $\mu$ by the equation

[TABLE]

where $\lceil r\rceil$ denotes the least integer greater than or equal to the real number $r$. By a result of Ax[4] (see also Katz[5, Proposition 2.4]) we have

[TABLE]

Our goal in this paper is to give a mod $p$ congruence for $P_{\lambda}(q^{-\mu}t)$. We do this by defining a generalization of the classical Hasse-Witt matrix, which gives such a congruence for $\mu=0$. Presumably our matrix is the matrix of a “higher Hasse-Witt” operation as defined by Katz[6, Section 2.3.4], but so far we have not been able to prove this.

It will be convenient to define an augmentation of the vectors ${\bf a}_{j}$. Set

[TABLE]

where ${\mathbb{N}}$ denotes the nonnegative integers. Note that the vectors ${\bf a}_{j}^{+}$ all lie on the hyperplane $\sum_{i=0}^{n}u_{i}=du_{n+1}$ in ${\mathbb{R}}^{n+2}$. We shall be interested in the lattice points on this hyperplane that lie in $({\mathbb{R}}_{>0})^{n+2}$: set

[TABLE]

Note that $u\in U$ implies that $u_{n+1}\geq\mu+1$. Let

[TABLE]

a nonempty set by the definition of $\mu$. We define a matrix of polynomials with rows and columns indexed by $U_{\min}$: let $A(\Lambda)=[A_{uv}(\Lambda)]_{u,v\in U_{\min}}$, where

[TABLE]

Note that since the $(n+1)$-st coordinate of each ${\bf a}_{j}^{+}$ equals 1, the condition on the summation implies that

[TABLE]

When $\mu=0$, it follows that $\nu_{j}\leq p-1$ for all $j$, hence the matrix $A(\Lambda)$ can be reduced modulo $p$. We denote by $\bar{A}(\Lambda)\in{\mathbb{F}}_{p}[\Lambda]$ its reduction modulo $p$. Using the algorithm of Katz[6, Algorithm 2.3.7.14], one then checks that $\bar{A}(\lambda)$ is the Hasse-Witt matrix of the hypersurface $f_{\lambda}=0$. It is somewhat surprising that even when $\mu>0$ we still have $\nu_{j}\leq p-1$ for all $j$.

Lemma 1.3**.**

If $u,v\in U_{\min}$, $\nu\in{\mathbb{N}}^{N}$, and $\sum_{j=1}^{N}\nu_{j}{\bf a}_{j}^{+}=pu-v$, then $\nu_{j}\leq p-1$ for all $j$. In particular, $A_{uv}(\Lambda)\in({\mathbb{Q}}\cap{\mathbb{Z}}_{p})[\Lambda]$, so $A_{uv}(\Lambda)$ can be reduced modulo $p$.

The proof of Lemma 1.3 will be given in Section 2. By the results of [1, Theorem 2.7 or Theorem 3.1], which will be recalled in Section 2, Lemma 1.3 implies immediately that each $A_{uv}(\Lambda)$ is a mod $p$ solution of an $A$-hypergeometric system of differential equations.

Write the rational function $P_{\lambda}(t)$ of (1.2) as

[TABLE]

where $Q_{\lambda}(t)$ and $R_{\lambda}(t)$ are relatively prime polynomials with

[TABLE]

If $X_{\lambda}$ is smooth, it is known that $P_{\lambda}(t)$ is a polynomial, i. e., $R_{\lambda}(t)=1$. Our main result is the following, which does not require any smoothness assumption.

Theorem 1.4**.**

If $n$ is not divisible by $d$, then $R_{\lambda}(q^{-\mu}t)\equiv 1\pmod{q}$ and

[TABLE]

Note that even in the classical case of the Hasse-Witt matrix ($\mu=0$), this result contains something new, as we do not assume that $X_{\lambda}$ is a smooth hypersurface.

The proof of Theorem 1.4 will occupy Sections 3–5. To describe the zeta function, we apply the $p$-adic cohomology theory of Dwork, as in Katz[7, Sections 4–6]. Indeed, Equation (3.5) below is a refined version of [7, Equation (4.5.33)]. We discuss the case $d\,|\,n$ in Section 6. If $d\,|\,n$, the conclusion of Theorem 1.4 need not hold, and the rational function $P_{\lambda}(q^{-\mu}t)\pmod{p}$ is instead described by Theorem 6.2. We prove the generic invertibility of the matrix $\bar{A}(\Lambda)$ in Section 7.

2. Proof of Lemma 1.3

It will be convenient for later applications to prove a more general version of Lemma 1.3. Put $S=\{0,1,\dots,n\}$ and let $I\subseteq S$. Define an integer $\mu_{I}$ by the equation

[TABLE]

Note that $\mu_{I}\geq 0$ if $I\neq\emptyset$, $\mu_{\emptyset}=-1$, and, in the notation of the Introduction, $\mu_{S}=\mu$. Set

[TABLE]

Note that $u\in U^{I}$ implies that $u_{n+1}\geq\mu_{I}+1$. Let

[TABLE]

a nonempty set by the definition of $\mu_{I}$. Lemma 1.3 is the special case $I=S$ of the following result.

Lemma 2.1**.**

If $u,v\in U^{I}_{\min}$, $\nu\in{\mathbb{N}}^{N}$, and $\sum_{j=1}^{N}\nu_{j}{\bf a}_{j}^{+}=pu-v$, then $\nu_{j}\leq p-1$ for all $j$.

Proof.

The result is trivial when $I=\emptyset$ since $U^{\emptyset}_{\min}=\{(0,\dots,0)\}$, so assume $I\neq\emptyset$. Let $u=(u_{0},\dots,u_{n},\mu_{I}+1),v=(v_{0},\dots,v_{n},\mu_{I}+1)\in U^{I}_{\min}$. Fix $k\in\{1,\dots,N\}$. We claim there exists an index $i_{0}\in\{0,\dots,n\}$ such that

[TABLE]

For if (2.2) fails for all $i_{0}\in\{0,\dots,n\}$, then

[TABLE]

contradicting the definition of $\mu_{I}$.

If $\nu_{k}\geq p$, then

[TABLE]

hence in both cases we have

[TABLE]

But our hypothesis $\sum_{j=1}^{N}\nu_{j}{\bf a}_{j}^{+}=pu-v$ implies that

[TABLE]

This contradiction shows that $\nu_{k}\leq p-1$. And since $k$ was arbitrary, the lemma is established. ∎

We recall the definition of the $A$-hypergeometric system of differential equations associated to the set $A=\{{\bf a}_{j}^{+}\}_{j=1}^{N}$. Let $L\subseteq{\mathbb{Z}}^{N}$ be the lattice of relations on $A$,

[TABLE]

and let $\beta=(\beta_{0},\dots,\beta_{n+1})\in{\mathbb{C}}^{n+2}$. The $A$-hypergeometric system with parameter $\beta$ is the system of partial differential operators in variables $\Lambda_{1},\dots,\Lambda_{N}$ consisting of the box operators

[TABLE]

and the Euler (or homogeneity) operators

[TABLE]

and

[TABLE]

Let $A^{I}(\Lambda)=[A^{I}_{uv}(\Lambda)]_{u,v\in U^{I}_{\min}}$, where

[TABLE]

Note that in the notation of the Introduction we have $A^{S}_{uv}(\Lambda)=A_{uv}(\Lambda)$. By Lemma 2.1, the polynomials $A^{I}_{uv}(\Lambda)$ have $p$-integral coefficients. Lemma 2.1 also says that $pu-v$ is very good in the sense of [1, Section 2]. We may therefore apply [1, Theorem 2.7] (or [1, Theorem 3.1] since this system is nonconfluent) to conclude that $\bar{A}^{I}_{uv}(\Lambda)$ is a mod $p$ solution of the $A$-hypergeometric system with parameter $\beta=pu-v$ (or, equivalently, $\beta=-v$ since we have reduced modulo $p$).

3. The zeta function

To make a connection between the matrix $A(\Lambda)$ and the zeta function (1.2), we apply a consequence of the Dwork trace formula developed in [3] (see Equation (3.5) below). Let $\gamma_{0}$ be a zero of the series $\sum_{i=0}^{\infty}t^{p^{i}}/p^{i}$ having ${\rm ord}_{p}\>\gamma_{0}=1/(p-1)$, where ${\rm ord}_{p}$ is the $p$-adic valuation normalized by ${\rm ord}_{p}\>p=1$. Let $L_{0}$ be the space of series

[TABLE]

For $I\subseteq\{0,\dots,n\}$, let $L_{0}^{I}$ be the subset of $L_{0}$ defined by

[TABLE]

Let ${\rm AH}(t)=\exp(\sum_{i=0}^{\infty}t^{p^{i}}/p^{i})$ be the Artin-Hasse series, a power series in $t$ that has $p$-integral coefficients, and set

[TABLE]

We then have

[TABLE]

We define the Frobenius operator on $L_{0}$. Put

[TABLE]

where $\hat{\lambda}$ denotes the Teichmüller lifting of $\lambda$. We shall also need to consider the series $\theta_{0}(\hat{\lambda},x)$ defined by

[TABLE]

Define an operator $\psi$ on formal power series by

[TABLE]

Denote by $\alpha_{\hat{\lambda}}$ the composition

[TABLE]

The map $\alpha_{\hat{\lambda}}$ operates on $L_{0}$ and is stable on each $L_{0}^{I}$. The proof of Theorem 1.4 will be based on the following formula for the rational function $P_{\lambda}(t)$ defined in (1.2). By [3, Equation 7.12] we have

[TABLE]

To exploit (3.5) we shall need $p$-adic estimates for the action of $\alpha_{\hat{\lambda}}$ on $L^{I}_{0}$. Expand (3.3) as a series in $x$, say,

[TABLE]

Note that from the definitions we have $\theta_{0,w}(\hat{\lambda})\in{\mathbb{Q}}_{p}(\zeta_{q-1},\gamma_{0})$. A direct calculation shows that for $v\in U^{I}$,

[TABLE]

thus we need $p$-adic estimates for the $\theta_{0,qu-v}(\hat{\lambda})$ with $u,v\in U^{I}$.

Expand (3.2) as a series in $x$:

[TABLE]

where

[TABLE]

with

[TABLE]

From (3.1) we have the estimate

[TABLE]

In particular, this implies the estimate

[TABLE]

By (3.3) and (3.8) we have

[TABLE]

In particular, we get the formula

[TABLE]

Applying (3.12) to the products on the right-hand side of (3.14) gives

[TABLE]

This estimate is not directly helpful for estimating $\theta_{0,qu-v}(\hat{\lambda})$ because we lack information about the $w^{(i)}$. Instead we proceed as follows.

Fix $w^{(0)},\dots,w^{(a-1)}\in{\mathbb{N}}A$ with

[TABLE]

We construct inductively from $\{w^{(i)}\}_{i=0}^{a-1}$ a related sequence $\{\tilde{w}^{(i)}\}_{i=0}^{a}\subseteq U^{I}$ such that

[TABLE]

First of all, take $\tilde{w}^{(0)}=v$. Eq. (3.16) shows that $w^{(0)}+\tilde{w}^{(0)}=p\tilde{w}^{(1)}$ for some $\tilde{w}^{(1)}\in{\mathbb{Z}}^{n+2}$; since $w^{(0)}\in{\mathbb{N}}A$ and $\tilde{w}^{(0)}\in U^{I}$ we conclude that $\tilde{w}^{(1)}\in U^{I}$. Suppose that for some $0<k\leq a-1$ we have defined $\tilde{w}^{(0)},\dots,\tilde{w}^{(k)}\in U^{I}$ satisfying (3.17) for $i=0,\dots,k-1$. Substituting $p\tilde{w}^{(i+1)}-\tilde{w}^{(i)}$ for $w^{(i)}$ for $i=0,\dots,k-1$ in (3.16) gives

[TABLE]

Since $\tilde{w}^{(0)}=v$, we can divide this equation by $p^{k}$ to get $\tilde{w}^{(k)}+w^{(k)}=p\tilde{w}^{(k+1)}$ for some $\tilde{w}^{(k+1)}\in{\mathbb{Z}}^{n+2}$. Since $w^{(k)}\in{\mathbb{N}}A$ and (by induction) $\tilde{w}^{(k)}\in U^{I}$, we conclude that $\tilde{w}^{(k+1)}\in U^{I}$. This completes the inductive construction. Note that in the special case $k=a-1$, this computation gives $\tilde{w}^{(a)}=u$.

Summing Eq. (3.17) over $i=0,\dots,a-1$ and using $\tilde{w}^{(0)}=v$, $\tilde{w}^{(a)}=u$, gives

[TABLE]

hence

[TABLE]

For $w^{(0)},\dots,w^{(a-1)}$ as in (3.16), we thus get from (3.15)

[TABLE]

Since $\tilde{w}^{(i)}\in U^{I}$, we have

[TABLE]

From (3.21) and (3.22) we get the following result.

Lemma 3.23**.**

For $u,v\in U^{I}$ and $w^{(0)},\dots,w^{(a-1)}$ as in $(3.16)$, we have

[TABLE]

Furthermore, if any of the terms $\tilde{w}^{(1)},\dots,\tilde{w}^{(a-1)}$ of the associated sequence satisfying $(3.17)$ is not contained in $U^{I}_{\min}$, then

[TABLE]

Our desired estimate for $\theta_{0,qu-v}(\hat{\lambda})$ now follows from (3.14).

Corollary 3.26**.**

For $u,v\in U^{I}$ we have

[TABLE]

4. The action of $\alpha_{\hat{\lambda}}$ on $L_{0}^{I}$

In this section, we use Corollary 3.26 to study the action of $\alpha_{\hat{\lambda}}$ on $L_{0}^{I}$. From (3.7) and the formula of Serre[8, Proposition 7] we have

[TABLE]

where

[TABLE]

the outer sum is over all subsets $U_{m}\subseteq U^{I}$ of cardinality $m$, and ${\mathcal{S}}_{m}$ is the group of permutations on $m$ objects.

Proposition 4.3**.**

The coefficient $a^{I}_{m}$ is divisible by $q^{m(\mu_{I}+1)}$ and satisfies the congruence

[TABLE]

In particular, $a^{I}_{m}\equiv 0\pmod{pq^{m(\mu_{I}+1)}}$ if $m>|U^{I}_{\min}|$.

Proof.

If $U_{m}\subseteq U^{I}$ is a subset of cardinality $m$ and $\sigma$ is a permutation of $U_{m}$, then by (3.27)

[TABLE]

since $u\in U_{m}$ implies $u_{n+1}\geq\mu_{I}+1$. It follows from (4.2) that $a^{I}_{m}$ is divisible by $q^{m(\mu_{I}+1)}$. Furthermore, $u_{n+1}\geq\mu_{I}+2$ if $u\not\in U^{I}_{\min}$, so

[TABLE]

The congruence (4.4) now follows from (4.2). ∎

As an immediate corollary of Proposition 4.3, we have the following result.

Corollary 4.5**.**

The reciprocal roots of $\det(I-t\alpha_{\hat{\lambda}}\mid L_{0}^{I})$ are all divisible by $q^{\mu_{I}+1}$.

Corollary 4.5 allows us to analyze the terms on the right-hand side of (3.5).

Proposition 4.6**.**

The reciprocal roots of $\det(I-q^{n+1-|I|}t\alpha_{\hat{\lambda}}\mid L_{0}^{I})$ are divisible by $q^{\mu+2}$ unless either $|I|=n+1$ or $|I|=n$ and $n$ is divisible by $d$, in which case they are divisible by $q^{\mu+1}$.

Proof.

Corollary 4.5 and the definition of $\mu_{I}$ imply that the reciprocal roots of $\det(I-q^{n+1-|I|}t\alpha_{\hat{\lambda}}\mid L_{0}^{I})$ are divisible by $q$ to the power

[TABLE]

If $|I|=n+1$, this reduces to $\mu+1$. Suppose $|I|=n$. From the definition of $\mu$ we have $n=\mu d+r$ with $0\leq r\leq d-1$. The expression (4.7) then reduces to $1+\lceil(\mu d+r)/d\rceil$, which equals $\mu+2$ if $r>0$ and equals $\mu+1$ if $r=0$.

If $|I|=n-1$, expression (4.7) reduces to $2+\lceil(\mu d+r-1)/d\rceil$, which equals $\mu+3$ if $r>1$ and equals $\mu+2$ if $r=0,1$. Finally, note that expression (4.7) cannot decrease when $|I|$ decreases, so expression (4.7) will be $\geq\mu+2$ for $|I|<n-1$. ∎

From (3.5) and Proposition 4.6 we get the following result.

Proposition 4.8**.**

If $n$ is not divisible by $d$, then

[TABLE]

If $n$ is divisible by $d$, then

[TABLE]

5. Proof of Theorem 1.4

It follows from (4.9) that $R_{\lambda}(q^{-\mu}t)\equiv 1\pmod{q}$ if $n$ is not divisible by $d$. To establish Theorem 1.4, it remains to prove the congruence for $Q_{\lambda}(q^{-\mu}t)$.

Consider the matrix $B^{I}(\hat{\lambda})=[B^{I}_{uv}(\hat{\lambda})]_{u,v\in U^{I}_{\min}}$ defined by

[TABLE]

We have

[TABLE]

where

[TABLE]

Proposition 4.3 implies $b_{m}^{I}\equiv 0\pmod{q^{m(\mu_{I}+1)}}$ and

[TABLE]

which establishes the following result.

Proposition 5.3**.**

We have

[TABLE]

Combined with Proposition 4.8, this gives the following congruences.

Corollary 5.4**.**

If $n$ is not divisible by $d$, then

[TABLE]

If $n$ is divisible by $d$, then

[TABLE]

To simplify (5.5) and (5.6) further, we restate Lemma 3.23 in the special case where $u,v\in U^{I}_{\min}$, i.e., $u_{n+1}=v_{n+1}=\mu_{I}+1$.

Lemma 5.7**.**

For $u,v\in U^{I}_{\min}$ and $w^{(0)},\dots,w^{(a-1)}$ as in $(3.16)$, we have

[TABLE]

Furthermore, if any of the terms $\tilde{w}^{(1)},\dots,\tilde{w}^{(a-1)}$ of the associated sequence satisfying $(3.17)$ is not contained in $U^{I}_{\min}$, then

[TABLE]

Applying Lemma 5.7 to Equations (3.14) and (3.17) gives the following congruence: for $u,v\in U^{I}_{\min}$,

[TABLE]

where $\tilde{w}^{(0)}=v$ and $\tilde{w}^{(a)}=u$.

Let $C^{I}(\hat{\lambda})=[C^{I}_{uv}(\hat{\lambda})]_{u,v\in U_{I,\min}}$ be the matrix

[TABLE]

It is straightforward to check by induction on $a$ that the right-hand side of (5.10) is the $(u,v)$-entry in the matrix product

[TABLE]

i.e., (5.10) implies the matrix congruence

[TABLE]

We make explicit the matrix $C^{I}(\hat{\lambda})$. Let $u,v\in U^{I}_{\min}$. From (3.9) and (3.10) we have

[TABLE]

Since $u,v\in U^{I}_{\min}$, Lemma 2.1 implies that $\nu_{j}\leq p-1$ for all $j$ in the sum on the right-hand side of (5.12). It then follows from the definition of $\theta(t)$ that $\theta_{\nu_{j}}=\gamma_{0}^{\nu_{j}}/\nu_{j}!$. Examining the last coordinate of the equation $\sum_{j=1}^{N}\nu_{j}{\bf a}_{j}^{+}=pu-v$ gives $\sum_{j=1}^{N}\nu_{j}=(p-1)(\mu_{I}+1)$, so (5.12) can be simplified to

[TABLE]

Using (2.3), we obtain the relation between the matrices $C^{I}(\hat{\lambda})$ and $A^{I}(\Lambda)$:

[TABLE]

From (5.11), we then obtain a relation between $A^{I}(\hat{\lambda})$ and $B^{I}(\hat{\lambda})$:

[TABLE]

Recall that $\sum_{i=0}^{\infty}\gamma_{0}^{p^{i}}/p^{i}=0$ and that ${\rm ord}_{p}\>\gamma_{0}=1/p-1$. In particular,

[TABLE]

and since the right-hand side of this expression is an increasing function of $i$ for $i\geq 1$ we have

[TABLE]

Multiplying by $p/\gamma_{0}$ then gives

[TABLE]

It follows from this that

[TABLE]

so (5.15) may be simplified to

[TABLE]

Corollary 5.4 now implies the following congruences.

Theorem 5.17**.**

If $n$ is not divisible by $d$, then

[TABLE]

If $n$ is divisible by $d$, then

[TABLE]

Since $A^{S}(\Lambda)$ is the matrix denoted $A(\Lambda)$ in the Introduction and since $\hat{\lambda}$ is the Teichmüller lifting of $\lambda$ we have $A^{S}(\hat{\lambda})\equiv\bar{A}^{S}(\lambda)\pmod{p}$. Theorem 1.4 now follows from (5.18), and (5.19) is equivalent to

[TABLE]

6. The case $d\,|\,n$

We first give an example to show that Theorem 1.4 fails when $d\,|\,n$. Consider the variety $X$ in ${\mathbb{P}}^{n}$ defined by the equation $x_{1}\cdots x_{n}=0$ (so we have $d=n$ and $\mu=1$). A short calculation shows that its zeta function $Z(X,t)$ has the form (1.2) with

[TABLE]

In particular, we have $R(q^{-1}t)\equiv(1-t)\pmod{q}$, contradicting Theorem 1.4.

An elementary observation will allow us to simplify the denominator of (5.20). Assume that $d\,|\,n$ and $i\in\{0,1,\dots,n\}$. Then

[TABLE]

This implies that $U^{S\setminus\{i\}}_{\min}$ is a singleton: $U^{S\setminus\{i\}}_{\min}=\{u^{(i)}\}$, where

[TABLE]

with zero in the $i$-th entry. It follows that if we define a polynomial $g_{i}(\Lambda)\in({\mathbb{Q}}\cap{\mathbb{Z}}_{p})[\Lambda]$ by the formula

[TABLE]

then by (2.3), $A^{S\setminus\{i\}}(\Lambda)$ is a one-by-one matrix with entry $g_{i}(\Lambda)$. From (5.20) we then have the following result.

Theorem 6.2**.**

If $d\,|\,n$, then

[TABLE]

7. Generic invertibility of $\bar{A}^{I}(\Lambda)$

The proof of generic invertibility follows the lines of our recent proof of generic invertibility for the Hasse-Witt matrix[2]. We fix $I$ and prove that the matrix $\bar{A}^{I}(\Lambda)$ is generically invertible, in a sense made precise below.

We consider the following condition on the set $\{{\bf a}_{j}\}_{j=1}^{N}$. Suppose that $N\geq\mu_{I}+|U^{I}_{\min}|$ and that the elements $\{{\bf a}_{j}\}_{j=1}^{\mu_{I}}$ and $\{{\bf a}_{j}\}_{j=\mu_{I}+1}^{\mu_{I}+|U^{I}_{\min}|}$ have the following property:

Hypothesis 7.1**.**

For each $u\in U^{I}_{\min}$, there exists (a necessarily unique) $k_{u}$, $1\leq k_{u}\leq|U^{I}_{\min}|$, such that $x^{u}=x^{{\bf a}_{\mu_{I}+k_{u}}^{+}}\prod_{j=1}^{\mu_{I}}x^{{\bf a}_{j}^{+}}$.

We show that such subsets always exists when, for example, $\{x^{{\bf a}_{j}}\}_{j=1}^{N}$ consists of all monomials of degree $d$. To fix ideas and simplify notation, suppose that $|I|=h$, $0\leq h\leq n+1$ and that $I=\{0,1,\dots,h-1\}$. For $j=1,\dots,\mu_{I}$ we take $x^{{\bf a}_{j}}=\prod_{k=(j-1)d}^{jd-1}x_{k}$ and we take $\{x^{{\bf a}_{j}}\}_{j=\mu_{I}+1}^{\mu_{I}+|U^{I}_{\min}|}$ to consist of all monomials of degree $d$ that are divisible by the product $x_{\mu_{I}d}\cdots x_{h-1}$. Then Hypothesis 7.1 is satisfied.

Theorem 7.2**.**

If $\{{\bf a}_{j}\}_{j=1}^{N}$ satisfies Hypothesis $7.1$, then the matrix $\bar{A}^{I}(\Lambda)$ is invertible.

We begin with a reduction step. Define a related matrix $D^{I}(\Lambda)=[D^{I}_{uv}(\Lambda)]_{u,v\in U^{I}_{\min}}$ by setting

[TABLE]

where $A^{I}_{uv}(\Lambda)$ is given by (2.3). In other words, we obtain $D^{I}(\Lambda)$ by multiplying row $u$ of $A^{I}(\Lambda)$ by \big{(}\Lambda_{\mu_{I}+k_{u}}\prod_{j=1}^{\mu_{I}}\Lambda_{j}\big{)}^{-p} and multiplying column $v$ of $A^{I}(\Lambda)$ by $\Lambda_{\mu_{I}+k_{v}}\prod_{j=1}^{\mu_{I}}\Lambda_{j}$. It follows that

[TABLE]

hence to prove Theorem 7.2 it suffices to prove that

[TABLE]

For $u\in U^{I}_{\min}$, put

[TABLE]

Lemma 7.5**.**

If the monomial $\Lambda^{l}$ appears in the Laurent polynomial $D^{I}_{uv}(\Lambda)$, then $l\in L_{u}$.

Proof.

It follows from Hypothesis 7.1 that

[TABLE]

The formula for $A^{I}_{uv}(\Lambda)$ is given in (2.3), where each exponent $\nu_{j}$ satisfies $0\leq\nu_{j}\leq p-1$ by Lemma 2.1. The assertion of Lemma 7.5 now follows from the definition of $D^{I}_{uv}(\Lambda)$. ∎

Lemma 7.6**.**

The Laurent polynomial $D^{I}_{uv}(\Lambda)$ has no constant term if $u\neq v$ and has constant term equal to \big{(}-(p-1)!\big{)}^{-(\mu_{I}+1)} if $u=v$.

Proof.

If $u\neq v$, the definition of $D^{I}_{uv}(\Lambda)$ shows that every monomial in $D^{I}_{uv}(\Lambda)$ contains a negative power of $\Lambda_{\mu_{I}+k_{u}}$ and a positive power of $\Lambda_{\mu_{I}+k_{v}}$. If $u=v$, then

[TABLE]

so (2.3) shows that the monomial

[TABLE]

appears in $A^{I}_{uv}(\Lambda)$. The assertion of the lemma then follows from the definition of $D^{I}_{uv}(\Lambda)$. ∎

We shall prove (7.4) by showing that $\det\bar{D}^{I}(\Lambda)$ has a nonzero constant term. In fact, we shall show that the constant term of $\det D^{I}(\Lambda)$ is a $p$-adic unit. By Lemma 7.5 and the following proposition, the constant term of $\det D^{I}(\Lambda)$ is the determinant of the matrix whose $(u,v)$-entry is the constant term of $D^{I}_{uv}(\Lambda)$. And by Lemma 7.6, this matrix of constant terms is a diagonal matrix whose diagonal entries are $p$-adic units. Thus the following proposition completes the proof of (7.4).

Proposition 7.7**.**

Let $l^{(u)}=(l^{(u)}_{1},\dots,l^{(u)}_{N})\in L_{u}$ for $u\in U^{I}_{\min}$. One has

[TABLE]

if and only if $l^{(u)}={\bf 0}$ for all $u\in U^{I}_{\min}$.

Proof.

The “if” part of the proposition is clear, so consider a set $\{l^{(u)}\}_{u\in U^{I}_{\min}}$ with $l^{(u)}\in L_{u}$ satisfying (7.8). Since $l^{(u)}\in L_{u}$ we have

[TABLE]

which implies (since the last coordinate of each ${\bf a}_{k}^{+}$ equals 1)

[TABLE]

From the definition of $L_{u}$, we have $l^{(u)}_{j}\leq 0$ for all $u$ if $j\leq\mu_{I}$ and $l^{(u)}_{j}\geq 0$ for all $u$ if $j\geq\mu_{I}+|U^{I}_{\min}|+1$, so (7.8) implies that

[TABLE]

and

[TABLE]

To establish the proposition, it remains to show that $l^{(u)}_{j}=0$ for all $u$ if $\mu_{I}+1\leq j\leq\mu_{I}+|U^{I}_{\min}|$.

Using (7.11) and (7.12), Equations (7.9) and (7.10) become

[TABLE]

and

[TABLE]

for all $u$. Since $l^{(u)}\in L_{u}$, we have $l^{(u)}_{\mu_{I}+k_{u}}\leq 0$ and $l^{(u)}_{\mu_{I}+k}\geq 0$ if $k\neq k_{u}$. If $l^{(u)}_{\mu_{I}+k_{u}}=0$, then (7.14) implies that $l^{(u)}_{\mu_{I}+k}=0$ for $k\neq k_{u}$ as well, so $l^{(u)}={\bf 0}$.

If follows that if $l^{(u)}\neq{\bf 0}$, then $l^{(u)}_{\mu_{I}+k_{u}}<0$ so we can solve (7.13) for ${\bf a}_{k_{u}}^{+}$:

[TABLE]

The coefficients on the right-hand side of (7.15) are nonnegative rational numbers that sum to 1, so (7.15) implies the following statement.

Lemma 7.16**.**

If $l^{(u)}\neq{\bf 0}$, then ${\bf a}_{k_{u}}^{+}$ is an interior point of the convex hull of the set $\{{\bf a}_{k}^{+}\mid l^{(u)}_{\mu_{I}+k}>0\}$.

Let Z=\{{\bf a}_{k}^{+}\mid\text{l^{(u)}{\mu{I}+k}\neq 0$for some$u\in U^{I}_{\min}}\}. If ${\bf a}_{k}^{+}\not\in Z$, then $l^{(u)}_{\mu_{I}+k}=0$ for all $u\in U^{I}_{\min}$, so if $Z=\emptyset$ then $l^{(u)}_{\mu_{I}+k}=0$ for all $k$, $1\leq k\leq|U^{I}_{\min}|$, and all $u\in U^{I}_{\min}$, which establishes the proposition. So suppose $Z\neq\emptyset$ and choose $k_{0}$, $1\leq k_{0}\leq|U^{I}_{\min}|$, such that ${\bf a}_{k_{0}}^{+}$ is a vertex of the convex hull of $Z$. If $l^{(u)}\neq{\bf 0}$, then Lemma 7.16 implies that $k_{0}\neq k_{u}$. By the definition of $L_{u}$, we therefore have $l^{(u)}_{\mu_{I}+k_{0}}\geq 0$ for all $u\in U^{I}_{\min}$. Furthermore, since ${\bf a}_{k_{0}}^{+}\in Z$, we must have $l^{(u)}_{\mu_{I}+k_{0}}>0$ for some $u\in U^{I}_{\min}$. It follows that $\sum_{u\in U^{I}_{\min}}l^{(u)}_{\mu_{I}+k_{0}}>0$, contradicting (7.8). Thus we must have $Z=\emptyset$. ∎