Local Galois Symbols on E × E

This article studies the Albanese kernel T_F(E x E), for an elliptic curve E over a p-adic field 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 x E)/p to the Galois cohomology group H^2 (F,E[P] ⊗ E[P]), for E/F ordinary, without requiring that the p-torsion points are F-rational, or even that the Galois module E[P] is semisimple. A key step is to show that the image is zero when the finite Galois module E[P] is acted on non-trivially by the pro-p-inertia group I_p. Non-trivial classes in the image are also constructed when E[P] is suitably unramified. A forthcoming sequel will deal with global questions.