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2021 $p$-adic estimates of exponential sums on curves
Joe Kramer-Miller
Algebra Number Theory 15(1): 141-171 (2021). 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 𝔽q of characteristic p3 and let VX 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 xXV. 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.

Citation

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Joe Kramer-Miller. "$p$-adic estimates of exponential sums on curves." Algebra Number Theory 15 (1) 141 - 171, 2021. https://doi.org/10.2140/ant.2021.15.141

Information

Received: 10 October 2019; Revised: 26 May 2020; Accepted: 24 June 2020; Published: 2021
First available in Project Euclid: 17 March 2021

Digital Object Identifier: 10.2140/ant.2021.15.141

Subjects:
Primary: 14F30
Secondary: 11G20, 11T23

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.15 • No. 1 • 2021
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