Robba’s method on Exponential sums

The basic objects of this study are exponential sums on a torus of dimension $n$ defined over a finite field $k$ with $\textrm{char}(k)=p$. Our methods are based on the work of Dwork, Adolphson and Sperber. In [4], Robba gives an explicit calculation of one variable twisted exponential sums. In fact, his method can be applied to the case of multi-variables.

Let $\zeta_{p}$ be a primitive $p$-th root of unity. Let $\psi$ be the additive character of $k$ given by $\psi(t)=\zeta_{p}^{\operatorname{Tr}_{k/\mathbf{F}_{p}}(t)}$. Let $f$ be a Laurent polynomial and write

$$f=\sum_{i=1}^{N}a_{i}x^{w_{i}}\in k[x_{1},\cdots,x_{n},x_{1}^{-1},\cdots,x_{n}^{-1}].$$

We assume that $a_{i}\neq 0$ for all $i$. Define exponential sums

$$S_{i}(f)=\sum_{x\in\mathbf{T}^{n}(k_{i})}\psi(\operatorname{Tr}_{k_{i}/k}(f(x))),$$

where $k_{i}$ are the extensions of $k$ of degree $i$. The $L$-function is defined by

$$L(f,t)=\exp\Big{(}\sum_{i=1}^{\infty}S_{i}(f)t^{i}/i\Big{)}.$$

In [1, section 2], Adolphson and Sperber use Dwork’s method to prove that $L(f,t)^{(-1)^{n-1}}$ is a polynomial when $f$ is nondegenerate. Moreover, they give a low bound of the Newton polygon of $L(f,t)^{(-1)^{n-1}}$ in [1, section 3], which we call Hodge polygon in this article. In our study, we want to give a more precise result about the Newton polygon when $f$ has only $n$ terms, that is $N=n$. Note that if we assume that $J=(w_{1},\cdots,w_{n})$ is invertible in $\mathbf{M}_{n}(\mathbf{R})$, we can found a solution $b=(b_{1},\cdots,b_{n})\in\bar{k}^{\times}$ such that $a_{i}b^{w_{i}}=1$ for all $i$. From now on, we assume that $(p,\det J)=1$, $k=\mathbf{F}_{p}$ and

$$f=\sum_{i=1}^{n}x^{w_{i}}.$$

Let $\Delta(f)$ be the Newton polyhedron at $\infty$ of $f$ which is defined to be the convex hull in $\mathbf{R}^{n}$ of the set $\left\{w_{j}\right\}_{j=1}^{n}\cup\left\{(0,\cdots,0)\right\}$ and let $C(f)$ be the convex cone generated by $\left\{w_{j}\right\}_{j=1}^{n}$ in $\mathbf{R}^{n}$. Let $\operatorname{Vol}(\Delta(f))$ be the volume of $\Delta(f)$ with respect to Lebesgue measure on $\mathbf{R}^{n}$. We say $f$ is nondegenerate with respect to $\Delta(f)$ if for any face $\sigma$ of $\Delta(f)$ not containing the origin, the Laurent polynomials $\frac{\partial f_{\sigma}}{\partial x_{i}}$, $i=1,\cdots,n$ have no common zero in $(\bar{k}^{\times})^{n}$, where $f_{\sigma}=\sum_{w_{j}\in\sigma}a_{j}x^{w_{j}}$. Set $M(f)=C(f)\cap\mathbf{Z}^{n}$. Note that $(p,\det J)=1$ implies that $f$ is nondegenerate. Since we have assumed that $J$ is invertible, any element $u\in M(f)$ can be uniquely written

(1.1)

$$u=\sum_{i=1}^{n}r_{i}w_{i}.$$

We define a weight on $M(f)$

$$w(u):=\sum_{i=1}^{n}r_{i}.$$

Note that the set of all elements $u\in M(f)$ such that all $0\leq r_{i}<1$ in the expression (1.1) form a fundamental domain of the lattice $M(f)$. Denote it by $S(\Delta)$. Note that $\operatorname{card}(S(\Delta))=n!\operatorname{Vol}{\Delta(f)}=\det(J)$ and $(p,\det J)=1$ imply that $S(\Delta)$ has a natural $p$-action. For any $u=r_{1}w_{1}+\cdots+r_{n}w_{n}\in S(\Delta)$, define

$$p.u=\sum_{i=1}^{n}\left\{pr_{i}\right\}w_{i},$$

where $\left\{pr_{i}\right\}$ is the fractional part of $pr_{i}$ for each $i$. We say $S(\Delta)$ is $p$-stable under weight function if $w(u)=w(p.u)$ for any $u\in S(\Delta)$. Now we give our main result.

Wan uses the Gauss sum to give an explicit formula of the $L$-function for the diagonal Laurent polynomial. Then he uses Stickelberger’s theorem to give a proof of above theorem. See [5, Theorem 3.4]. In this article, we use Robba’s method to prove above theorem. Indeed, Robba’s method can also be applied to prove [1, Theorem 3.10] and it is easier than the method used in [1, §3].

Let $\mathbf{Q}_{p}$ be the $p$-adic numbers. Let $\Omega$ be the completion of the algebraic closure of $\mathbf{Q}_{p}$. Denote by “ord” the additive valuation on $\Omega$ normalized by $\operatorname{ord}(p)=1$. The norm on $\Omega$ is given by $|u|=p^{-\operatorname{ord}(u)}$ for any $u\in\Omega$.

Note that there is an integer $M$ such that $w(M(f))\subset\frac{1}{M}\mathbf{Z}$. In [1, section 1], Adolphson and Sperber introduce a filtration on $R(f):=k[x^{M(f)}]$ given by

$$R(f)_{i/M}=\left\{\sum_{u\in M(f)}b_{u}x^{u}|w(u)\leq i/M~{}\textrm{for all $u$ with}~{}b_{u}\neq 0\right\}.$$

The associated graded ring is

$$\bar{R}=\bigoplus_{i\in\mathbf{Z}_{\geq 0}}\bar{R}^{i/M},$$

where

$$\bar{R}^{i/M}=R(f)_{i/M}/R(f)_{(i-1)/M}.$$

For $1\leq i\leq n$, let $\bar{f}_{i}$ be the image of $x_{i}\frac{\partial f}{\partial x_{i}}\in R(f)_{1}$ in $\in\bar{R}^{1}$. Let $\bar{I}$ be the ideal generated by $\bar{f}_{1},\dots,\bar{f}_{n}$ in $\bar{R}$. By [1, Theorem 2.14] and [1, Theorem 2.18], $\bar{f}_{1},\dots,\bar{f}_{n}$ in $\bar{R}$ form a regular sequence in $\bar{R}$ and $\dim_{k}\bar{R}/\bar{I}=n!\operatorname{Vol}(\Delta(f))$. For each integer $i$, we have a decomposition

(2.1)

$$\bar{R}^{i/M}=\bar{V}^{i/M}\oplus(\bar{R}^{i/M}\cap\bar{I}).$$

Set $a_{i}$=$\dim_{k}\bar{V}^{i/M}$.

For a non-negative integer $l$, set

$$W(l)=\operatorname{card}\left\{u\in M(f)|w(u)=\frac{l}{M}\right\}.$$

Note that this is a finite number for each $l$. Set

$$H(i)=\sum_{l=0}^{n}(-1)^{l}\binom{n}{l}W(i-lM).$$

Note that $\bar{R}/\bar{I}$ has a finite basis $S=\left\{x^{u}|u\in S(\Delta)\right\}$ and $\operatorname{card}(S)=n!\operatorname{Vol}(\Delta(f))$.

Consider the Artin-Hasse exponential series: $E(t)=\exp\Big{(}\sum_{i=0}^{\infty}\frac{t^{p^{i}}}{p^{i}}\Big{)}.$ By [2, Lemma 4.1], the series $\sum_{i=0}^{\infty}\frac{t^{p^{i}}}{p^{i}}$ has a zero at $\gamma\in\Omega$ such that $\operatorname{ord}\gamma=1/(p-1)$ and $\zeta_{p}\equiv 1+\gamma\mod\gamma^{2}$. Set

$$\theta(t)=E(\gamma t)=\sum_{i=0}^{\infty}c_{i}t^{i}.$$

The series $\theta(t)$ is a splitting function in Dwork’s terminology [2, §4a]. In particular, we have $\operatorname{ord}c_{i}\geq i/(p-1)$, $\theta(t)\in\mathbf{Q}_{p}(\zeta_{p})[[t]]$ and $\theta(1)=\zeta_{p}$. Fix an $M$-th root $\widetilde{\gamma}$ of $\gamma$ in $\Omega$. Let $K=\mathbf{Q}_{p}(\widetilde{\gamma})$, and $\mathcal{O}_{K}$ the ring of integers of $K$. Let $\hat{a}_{j}\in K$ be the Techmüller lifting of $a_{j}$ and set

$$\hat{f}(x)=\sum_{j=1}^{N}\hat{a}_{j}x^{\omega_{j}}\in K[x_{1},x_{1}^{-1},\cdots,x_{n},x_{n}^{-1}].$$

Consider the following spaces of $p$-adic functions

$$B_{0}=\left\{\sum_{u\in M(f)}A_{u}\widetilde{\gamma}^{Mw(u)}x^{u}|A_{u}\in\mathcal{O}_{K},A_{u}\rightarrow 0~{}\textrm{as}~{}u\rightarrow 0\right\},$$

$$B=\left\{\sum_{u\in M(f)}A_{u}\widetilde{\gamma}^{Mw(u)}x^{u}|A_{u}\in K,A_{u}\rightarrow 0~{}\textrm{as}~{}u\rightarrow 0\right\}.$$

Set $\gamma_{l}=\sum\limits_{i=0}^{l}\gamma^{p^{i}}/p^{i},h(t)=\sum\limits_{l=0}^{\infty}\gamma_{l}t^{p^{l}}$. Define

$$H(x)=\sum_{j=1}^{n}h(x^{w_{j}}),~{}F_{0}(x)=\prod_{i=1}^{n}\theta(x^{w_{i}})=\sum_{v\in M(f)}h_{v}x^{v}.$$

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

$$\psi(\sum_{u\in\mathbf{Z}^{n}}a_{u}x^{u})=\sum_{u\in\mathbf{Z}^{n}}a_{pu}x^{u}.$$

Let $\alpha=\psi\circ F_{0}(x)$. For $i=1,\cdots,n$, define operators

$$E_{i}=x_{i}\partial/\partial x_{i},~{}\hat{D}_{i}=E_{i}+E_{i}(H)$$

By [1, Corollary 2.9], we have

$\displaystyle L(f,t)^{(-1)^{n-1}}=\det(1-t\alpha|B/\sum_{i=1}^{n}\hat{D}_{i}B).$

By [1, Therorem 2.18, Theorem A.1], $S=\left\{x^{u}\right\}_{u\in S(\Delta)}$ is a free basis of $B/\sum_{i=1}^{n}\hat{D}_{i}B$. For any $u\in M(f),u^{\prime}\in S(\Delta)$, define $A(u,u^{\prime})$ by the relations

$$x^{u}\equiv\sum_{u^{\prime}\in S(\Delta)}A(u,u^{\prime})x^{u^{\prime}}\mod\sum_{i=1}^{n}\hat{D}_{i}B.$$

For any $u,u^{\prime}\in S(\Delta)$, define $\gamma(u,u^{\prime})$ by the relations

$$\alpha(x^{u})\equiv\sum_{u^{\prime}\in S(\Delta)}\gamma(u,u^{\prime})x^{u^{\prime}}\mod\sum_{i=1}^{n}\hat{D}_{i}B.$$

The main purpose is to give estimate for the $p$-adic valuations of the coefficients $\gamma(u,u^{\prime})$.

For any $u\in M(f)$, there is a unique $u^{\prime}\in S(\Delta)$ such that

$$u\in S_{u^{\prime}}=\left\{u^{\prime}+\sum_{i=1}^{n}\mathbf{Z}_{\geq 0}w_{i}\right\}.$$

Set $R_{u^{\prime}}=\left\{\xi=\sum a_{u}x^{u}\in B_{0}|u\in S_{u^{\prime}}\right\}$.