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

David Burns

Documenta mathematica, Tome 9 (2004), pp. 357-399

Résumé

Let K/k be a finite Galois extension of global function fields of characteristic p. Let CK​ 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 Gm​ on the corresponding open subschemes of CK​. 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 K0​(zl​[G],Ql​) 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.

Détail

DOI : 10.4171/dm/170

Classification : 11G40, 11R65, 19A31, 19B28