Vol. 15, No. 1, 2021

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$p$-adic estimates of exponential sums on curves

Joe Kramer-Miller

Vol. 15 (2021), No. 1, 141–171

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 𝔽q of characteristic p 3 and let V X be an affine curve. For a regular function f̄ on V , we may form the L-function L(f̄,V,s) associated to the exponential sums of f̄. In this article, we prove a lower estimate on the Newton polygon of L(f̄,V,s). The estimate depends on the local monodromy of f around each point x 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 p in terms of local monodromy invariants.

$p$-adic cohomology, Artin–Schreier covers, wild ramification, zeta function, Newton polygon, exponential sums
Mathematical Subject Classification 2010
Primary: 14F30
Secondary: 11G20, 11T23
Received: 10 October 2019
Revised: 26 May 2020
Accepted: 24 June 2020
Published: 1 March 2021
Joe Kramer-Miller
University of California Irvine
510V Rowland Hall
Ring Road
Irvine, CA 92612
United States