p-adic estimates of exponential sums on curves

Kramer-Miller, Joe

doi:10.2140/ant.2021.15.141

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$p$-adic estimates of exponential sums on curves

Joe Kramer-Miller

Vol. 15 (2021), No. 1, 141–171

DOI: 10.2140/ant.2021.15.141

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 $𝔽_{q}$ of characteristic $p \geq 3$ and let $V \subset X$ be an affine curve. For a regular function $\bar{f}$ on $V$ , we may form the $L$ -function $L (\bar{f}, V, s)$ associated to the exponential sums of $\bar{f}$ . In this article, we prove a lower estimate on the Newton polygon of $L (\bar{f}, V, s)$ . 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 $ℤ ∕ p ℤ$ in terms of local monodromy invariants.

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Keywords

$p$-adic cohomology, Artin–Schreier covers, wild ramification, zeta function, Newton polygon, exponential sums

Mathematical Subject Classification 2010

Primary: 14F30

Secondary: 11G20, 11T23

Milestones

Received: 10 October 2019

Revised: 26 May 2020

Accepted: 24 June 2020

Published: 1 March 2021

Authors

Joe Kramer-Miller
Mathematics University of California Irvine 510V Rowland Hall Ring Road Irvine, CA 92612 United States

Vol. 15, No. 1, 2021