$p$-adic estimates of exponential sums on curves

Abstract

The purpose of this article is to prove a “Newton over Hodge” result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field ${\mathbb{𝔽}}_q$ of characteristic $p\ge 3$ and let $V\subset X$ be an affine curve. For a regular function $\bar{f}$ on $V$, we may form the $L$-function $L\left(\bar{f},V,s\right)$ associated to the exponential sums of $\bar{f}$. In this article, we prove a lower estimate on the Newton polygon of $L\left(\bar{f},V,s\right)$. The estimate depends on the local monodromy of $f$ around each point $x\in X-V$. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on $p$-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $\mathbb{Z}∕p\mathbb{Z}$ in terms of local monodromy invariants.