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

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 $G:=\mathrm{Gal}(K/k)$. 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 ${\mathbb{G}}_m$ on the corresponding open subschemes of $C_K$. We then prove the $\ell$-primary component of this conjecture for all primes $\ell$ for which either $\ell \ne p$ or the relative algebraic $K$-group $K_0(z_{\ell }[G],{\mathbb{Q}}_{\ell })$ is torsion-free. In the remainder of the manuscript we show that this result has the following consequences for $K/k$: if $p\nmid |G|$, then refined versions of all of Chinburg's '$\Omega$-Conjectures' in Galois module theory are valid; if the torsion subgroup of $K^{\times }$ 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.