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

Abstract

This paper is a continuation to [Gaz17]. For every integer n≥1, we consider the generalized Galois symbol K(k;G1,G2)/n→snH2(k,G1[n]⊗G2[n]), where k is a finite extension of Qp, G1,G2 are semi-abelian varieties over k and K(k;G1,G2) is the Somekawa K-group attached to G1,G2. Under some mild assumptions, we describe the exact annihilator of the image of sn under the Tate duality perfect pairing, H2(k,G1[n]⊗G2[n])×H0(k,Hom(G1[n]⊗G2[n],μn))→Z/n. An important special case is when both G1,G2 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.