Global Galois symbols on [formula omitted]

Abstract

Let E be an elliptic curve over a number field F, A the abelian surface E×E, and TF(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 TF(A)/p in the Galois cohomology group H2(F,sym2(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 TF(A)/p is non-zero. Much of this work was joint with Jacob Murre, and the article is dedicated to his memory.