# A finer Tate duality theorem for local Galois symbols

## TL;DR

This paper extends Tate duality to a finer level for local Galois symbols associated with abelian varieties over p-adic fields, providing a geometric description of their images and their annihilators via p-adic Hodge theory.

## Contribution

It introduces a refined Tate duality theorem for Galois symbols related to abelian varieties, linking their images to integral p-adic Hodge theoretic objects.

## Key findings

- Provides a geometric description of the Galois symbol's image.
- Computes the annihilator of the Galois symbol's image under Tate duality.
- Connects the Galois symbol to integral p-adic Hodge theory.

## 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 $m\geq 1$, we consider the Galois symbol $K(K;A,B)/m\rightarrow H^2(K,A[m]\otimes B[m])$, where $K(K;A,B)$ is the Somekawa $K$-group attached to $A,B$. This map is a generalization of the Galois symbol $K_2^M(K)/m\rightarrow H^2(K,\mu_m^{\otimes 2})$ 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 $H^{2}(K,A[m]\otimes B[m])\times\mathrm{Hom}_{G_{K}}(A[m],B^{\star}[m])\rightarrow\mathbb{Z}/m,$ 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.