DOCUMENTA MATHEMATICA, Vol. 9 (2004), 357-399

David Burns

On the Values of Equivariant Zeta Functions of Curves over Finite Fields

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 $\bg_m$ 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 $\ell \not= p$ or the relative algebraic $K$-group $K_0(\bz _\ell [G],\bq_\ell)$ 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.

2000 Mathematics Subject Classification: Primary 11G40; Secondary 11R65 19A31 19B28

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