Global Galois Symbols on E x E

TL;DR

This paper investigates the Galois cohomology images of the Albanese kernel of E x E over number fields, revealing conditions under which these images vanish or are non-zero, with implications for elliptic curve arithmetic.

Contribution

It establishes new results on the behavior of the Albanese kernel modulo p for elliptic curves with ordinary reduction, including conditions for vanishing and non-vanishing of Galois cohomology images.

Findings

For all but finitely many primes p, the Galois cohomology image of T_F(A)/p is zero.

Existence of a finite extension K where T_K(A)/p is non-zero.

Results depend on the reduction type of E at prime p.

Abstract

Let E be an elliptic curve over a number field F, A the abelian surface E x E, and T_F(A) the F-rational albanese kernel of A, which is a subgroup of the degree zero part of Chow group of zero cycles on A modulo rational equivalence. The first result is that for all but a finite number of primes p where E has ordinary reduction, the image of T_F(A)/p in the Galois cohomology group H^2(F, sym^2(E[p])) is zero; here E[p] denotes as usual the Galois module of p-division points on E. The second result is that for any prime p where E has good ordinary reduction, there is a finite extension K of F, depending on p and E, such that T_K(A)/p is non-zero. Much of this work was joint with Jacob Murre, and the article is dedicated to his memory.