A Tate duality theorem for local Galois symbols II; The semi-abelian case

TL;DR

This paper extends Tate duality for Galois symbols associated with semi-abelian varieties over p-adic fields, describing the annihilator of the symbol's image and establishing finiteness results for related zero-cycles.

Contribution

It provides a description of the annihilator of Galois symbol images under Tate duality for semi-abelian varieties, including special cases with abelian varieties having split semistable reduction.

Findings

Explicit description of the annihilator of Galois symbol images

Finiteness results for zero-cycles on abelian varieties

Application to products of curves

Abstract

This paper is a continuation to \cite{Gazaki2017}. For every integer $n\geq 1$, we consider the generalized Galois symbol $K(k;G_1,G_2)/n\xrightarrow{s_n} H^2(k,G_1[n]\otimes G_2[n])$, where $k$ is a finite extension of $\mathbb{Q}_p$, $G_1,G_2$ are semi-abelian varieties over $k$ and $K(k;G_1,G_2)$ is the Somekawa K-group attached to $G_1, G_2$. Under some mild assumptions, we describe the exact annihilator of the image of $s_n$ under the Tate duality perfect pairing, $H^2(k,G_1[n]\otimes G_2[n])\times H^0(k,Hom(G_1[n]\otimes G_2[n],\mu_n))\rightarrow\mathbb{Z}/n$. An important special case is when both $G_1, G_2$ are abelian varieties with split semistable reduction. In this case we prove a finiteness result, which gives an application to zero-cycles on abelian varieties and products of curves.