Local Galois Symbols on E x E

TL;DR

This paper investigates the Galois symbol map on the Albanese kernel of the product of an elliptic curve over a p-adic field, revealing conditions under which the map's image is zero, without assuming rationality of p-torsion points.

Contribution

It provides new insights into the kernel and image of the Galois symbol map for E x E over p-adic fields, especially when E[p] is non-semisimple, without requiring rational p-torsion.

Findings

The Galois symbol map's image is zero when E[p] is non-semisimple.

The work applies to ordinary elliptic curves over p-adic fields.

It advances understanding of the Albanese kernel in this context.

Abstract

This article is the first part of a two-part work on the Albanese kernel T_F(E x E), for an elliptic curve E over F. The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from T_F(E \times E)/p to the Galois cohomology group H^2(F, E[p] (x) E[p]), for F a p-adic field and E/F ordinary, without requiring that the p-torsion points are F-rational. A key step is to show that the image is zero when the Galois module E[p] is non-semisimple. The forthcoming second part will deal with global questions.