On Tate duality for Jacobian varieties

Abstract

Let A be an abelian variety over a p-adic field k and A t its dual. The group of k-rational point A( k) has a p-adic decreasing filtration U · A( k). When A= J is a Jacobian variety, we give a precise description of the exact annihilator of U n A( k) with respect to the Tate pairing A(k)×H 1(k,A t)→ Q/ Z . As an application, we give another proof of the result of McCallum in the special case A= J, which says that U n A( k) annihilates ker( H 1( k, A t )→ H 1( k′, A t )) whenever k′/ k is a finite extension of conductor ⩽ n.