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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.20468
- arXiv:
- arXiv:2407.20468
- Bibcode:
- 2024arXiv240720468R
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Algebraic Geometry;
- 11G07;
- 14C25;
- 11G10
- E-Print:
- Twelve pages