Galois-module theory for wildly ramified covers of curves over finite fields

TL;DR

This paper extends Galois-module theory to wildly ramified covers of curves over finite fields, relating epsilon constants to Euler characteristics and generalizing previous results from tame to weakly ramified cases.

Contribution

It generalizes Chinburg's result from tame to weakly ramified covers and explores the integrality of epsilon constants in wildly ramified situations.

Findings

Relation between p-adic valuations of epsilon constants and Euler characteristics

Generalization of Chinburg's theorem to weakly ramified cases

Analysis of epsilon constants' integrality in wild ramification

Abstract

Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a `weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of $p$-adic valuations of epsilon constants.