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

TL;DR

This paper establishes a lower bound on the Newton polygon of abelian Artin L-functions on curves over finite fields, linking p-adic properties to monodromy invariants and confirming Deligne's predictions about irregular Hodge filtrations.

Contribution

It proves a 'Newton over Hodge' bound for L-functions of finite characters on curves, connecting p-adic estimates with monodromy and Hodge theory, and extends Deligne's conjectures.

Findings

Lower bounds on Newton polygons depend on Swan conductor and local exponents.

Under nondegeneracy, bounds match the irregular Hodge filtration.

Results imply p-adic bounds on L-functions for curves with cyclic symmetries.

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 $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $\rho:\pi_1^{et}(V) \to \mathbb{C}^\times$. In this article, we prove a lower bound on the Newton polygon of the $L$-function $L(\rho,s)$. The estimate depends on monodromy invariants of $\rho$: 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.