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

Let be a finite Galois extension of global function fields of characteristic . Let denote the smooth projective curve that has function field and set . We conjecture a formula for the leading term in the Taylor expansion at zero of the -equivariant truncated Artin -functions of in terms of the Weil-étale cohomology of on the corresponding open subschemes of . We then prove the -primary component of this conjecture for all primes for which either or the relative algebraic -group is torsion-free. In the remainder of the manuscript we show that this result has the following consequences for : if , then refined versions of all of Chinburg's '-Conjectures' in Galois module theory are valid; if the torsion subgroup of is a cohomologically-trivial -module, then Gross's conjectural 'refined class number formula' is valid; if 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