A finer Tate duality theorem for local Galois symbols

Abstract

Let $K$ be a finite extension of ${\mathbb{Q}}_p$. Let $A$, $B$ be abelian varieties over $K$ of good reduction. For any integer $m\ge 1$, we consider the Galois symbol $K(K;A,B)/m\to H^2(K,A[m]\otimes B[m])$, where $K(K;A,B)$ is the Somekawa $K$-group attached to $A,B$. This map is a generalization of the Galois symbol $K_2^M(K)/m\to H^2(K,{\mu }_m^{\otimes 2})$ of the Bloch-Kato conjecture, where $K_2^M(K)$ is the Milnor $K$-group of $K$. In this paper we give a geometric description of the image of this generalized Galois symbol by looking at the Tate duality pairing $H^2(K,A[m]\otimes B[m])\times {\mathrm{Hom}}_{G_K}(A[m],B^{\star }[m])\to \mathbb{Z}/m,$ where $B^{\star }$ is the dual abelian variety of $B$. Under this perfect pairing we compute the exact annihilator of the image of the Galois symbol in terms of an object of integral $p$-adic Hodge theory.

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