p-adic estimates of abelian Artin L-functions on curves

2020/06/08 by Joe Kramer-Miller, Kramer-Miller, Joe · 1 citation
Computer Science · Mathematics · #11G20 #11M38 #11T24 #14F30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2006.04936

openalex publication_date 2020/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let X be a smooth proper curve over a finite field \mathbbF_q of characteristic p≥ 3 and let V ⊂ X be an affine curve. Consider a nontrivial finite character ρ:π₁^et(V) → ℂ^×. In this article, we prove a lower bound on the Newton polygon of the L-function L(ρ,s). The estimate depends on monodromy invariants of ρ: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne's prediction that the irregular Hodge filtration would force p-adic bounds on L-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

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• Gauss--Heilbronn Sums and Coverings of Deligne--Lusztig Type Curves

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