# On the values of equivariant zeta functions of curves over finite fields

### David Burns

Dept. of Mathematics King's College London London WC2R 2LS United Kingdom

## Abstract

Let $K/k$ be a finite Galois extension of global function fields of characteristic $p$. Let $C_{K}$ denote the smooth projective curve that has function field $K$ and set $G:=Gal(K/k)$. We conjecture a formula for the leading term in the Taylor expansion at zero of the $G$-equivariant truncated Artin $L$-functions of $K/k$ in terms of the Weil-étale cohomology of $G_{m}$ on the corresponding open subschemes of $C_{K}$. We then prove the $ℓ$-primary component of this conjecture for all primes $ℓ$ for which either $ℓ=p$ or the relative algebraic $K$-group $K_{0}(z_{ℓ}[G],Q_{ℓ})$ is torsion-free. In the remainder of the manuscript we show that this result has the following consequences for $K/k$: if $p∤∣G∣$, then refined versions of all of Chinburg's '$Ω$-Conjectures' in Galois module theory are valid; if the torsion subgroup of $K_{×}$ is a cohomologically-trivial $G$-module, then Gross's conjectural 'refined class number formula' is valid; if $K/k$ satisfies a certain natural class-field theoretical condition, then Tate's recent refinement of Gross's conjecture is valid.

## Cite this article

David Burns, On the values of equivariant zeta functions of curves over finite fields. Doc. Math. 9 (2004), pp. 357–399

DOI 10.4171/DM/170