Weil-etale cohomology of curves over p-adic fields

Recent research has demonstrated a connection between Weil-kale cohomology and special values of zeta functions. In particular, Lichtenbaum has shown that the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field has a Weil-kale cohomological interpretation in terms of certain secondary Euler characteristics. These results rely on a duality theorem stated in terms of cup-product in Weil-kale cohomology. We define Weil-etale cohomology for varieties over p-adic fields, and prove a duality theorem for the cohomology of G(m) on a smooth, proper, geometrically connected curve of index 1. This duality theorem is a p-adic analogue of Lichtenbaum's Weil-kale duality theorem for curves over finite fields, as well as a Weil-kale analogue of his classical duality theorem for curves over p-adic fields. Finally, we show that our duality theorem implies this latter classical duality theorem for index 1 curves. (C) 2014 Elsevier Inc. All rights reserved.