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

Evangelia Gazaki

paper

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

arXiv:1808.05680

Abstract

This paper is a continuation to \cite{Gazaki2017}. For every integer $n \geq 1$ , we consider the generalized Galois symbol $K (k; G_{1}, G_{2}) / n s_{n} H^{2} (k, G_{1} [n] \otimes G_{2} [n])$ , where $k$ is a finite extension of $Q_{p}$ , $G_{1}, G_{2}$ 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, $H^{2} (k, G_{1} [n] \otimes G_{2} [n]) \times H^{0} (k, H o m (G_{1} [n] \otimes G_{2} [n], μ_{n})) \to Z / n$ . 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.

23 pages