Weil-etale Cohomology over Local Fields

Abstract

In a recent article, Lichtenbaum established the arithmetic utility of the Weil group of a finite field, by demonstrating a connection between certain Euler characteristics in Weil-etale cohomology and special values of zeta functions. In particular, the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field have a Weil-etale cohomological interpretation. These results rely on a duality theorem stated in terms of cup-product in Weil-etale cohomology.

With Lichtenbaum's paradigm in mind, we establish results for the cohomology of the Weil group of a local field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil group modules, which implies the main theorem of Local Class Field Theory. We define Weil- smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of G_m on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.