Global Galois Symbols on E x E
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.
Cite
@article{arxiv.2407.20468,
title = {Global Galois Symbols on E x E},
author = {Dinakar Ramakrishnan},
journal= {arXiv preprint arXiv:2407.20468},
year = {2024}
}
Comments
Twelve pages; this version has two added references, suggested by the referee, and a pair of minor typos have been fixed. The article has been accepted to appear in a special issue of Indagationes Mathematicae in memory of Jacob Murre, edited by Jan Nagel and Chris Peters