Galois symbol maps for abelian varieties over a $p$-adic field

TL;DR

This paper investigates the Galois symbol map for abelian varieties with good ordinary reduction over p-adic fields, providing insights into class groups via class field theory for curves.

Contribution

It introduces new methods to compute the Galois symbol map for abelian varieties over p-adic fields, linking it to class field theory.

Findings

Explicit calculation of the Galois symbol map in specific cases

Connection established between the symbol map and class groups

Enhanced understanding of abelian varieties over p-adic fields

Abstract

We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory for curves over a $p$-adic field.