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# Mathematical Research Letters

## Volume 15 (2008)

### Number 5

### Hodge-Stickelberger polygons for $L$-functions of exponential sums of $P(x^s)$

Pages: 1053 – 1071

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n5.a17

#### Authors

#### Abstract

Let $\F_q$ be a finite field of cardinality $q$ and characteristic $p$. Let $\bar{P}(x)$ be any one-variable Laurent polynomial over $\F_q$ of degree $(d_1,d_2)$ respectively and $p\nmid d_1d_2$. For any fixed $s\geq 1$ coprime to $p$, we prove that the $q$-adic Newton polygon of the $L$-functions of exponential sums of $\bar{P}(x^s)$ has a tight lower bound which we call Hodge-Stickelberger polygon, depending only on the $d_1,d_2,s$ and the residue class of $(p\bmod s)$. This Hodge-Stickelberger polygon is a certain weighted convolution of the Hodge polygon for $L$-function of exponential sums of $\bar{P}(x)$ and the Newton polygon for the $L$-function of exponential sums of $x^s$ (which is precisely given by the classical Stickelberger theory). We have an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. For any $\nu\in(\Z/s\Z)^\times$, we show that there exists a Zariski dense open subset $ {\cU} _\nu$ defined over $\Q$ such that for every Laurent polynomial $P$ in $\cU_\nu(\bar{\Q})$ the $q$-adic Newton polygon of $L(\bar{P}(x^s)/\F_q;T)$ converges to the Hodge-Stickelberger polygon as $p$ approaches infinity and $p\equiv \nu\bmod s$. As a corollary, we obtain a tight lower bound for the $q$-adic Newton polygon of the numerator of the zeta function of an Artin-Schreier curve given by affine equation $y^p-y=\bar{P}(x^s)$. This estimates the $q$-adic valuations of reciprocal roots of the numerator of the zeta function of the Artin-Schreier curve.