Weil-étale cohomology of curves over p-adic fields

Recent research has demonstrated a connection between Weil-étale 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-étale cohomological interpretation in terms of certain secondary Euler characteristics. These results rely on a duality theorem stated in terms of cup-product in Weil-étale cohomology.We define Weil-étale cohomology for varieties over p  -adic fields, and prove a duality theorem for the cohomology of GmGm on a smooth, proper, geometrically connected curve of index 1. This duality theorem is a p-adic analogue of Lichtenbaum's Weil-étale duality theorem for curves over finite fields, as well as a Weil-étale 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.