$p$-adic estimates of exponential sums on curves

TL;DR

This paper establishes lower bounds on the Newton polygon of exponential sums on algebraic curves over finite fields, linking $p$-adic valuations of Frobenius eigenvalues to local monodromy data, confirming a conjecture related to irregular Hodge filtrations.

Contribution

It proves a 'Newton over Hodge' estimate for exponential sums on curves, connecting local monodromy to $p$-adic valuations of Frobenius eigenvalues, advancing understanding of $L$-functions in arithmetic geometry.

Findings

Lower bounds on Newton polygons depending on local monodromy

Confirmation of Deligne's conjecture relating Hodge filtration and $p$-adic valuations

Application to curves with $bZ/pbZ$-action

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 $p\geq 3$ and let $V \subset X$ be an affine curve. For a regular function $\overline{f}$ on $V$, we may form the $L$-function $L(\overline{f},V,s)$ associated to the exponential sums of $\overline{f}$. In this article, we prove a lower estimate on the Newton polygon of $L(\overline{f},V,s)$. 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.