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-étale 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-étale cohomological interpretation. These results rely on a duality theorem stated in terms of cup-product in Weil-étale 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 Gm on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.
|Commitee:||Barg, Alexander, Haines, Thomas, Rosenberg, Jonathan, Washington, Lawrence|
|School:||University of Maryland, College Park|
|School Location:||United States -- Maryland|
|Source:||DAI-B 73/12(E), Dissertation Abstracts International|
|Keywords:||Algebraic geometry, Cohomology, Number theory, Weil groups|
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