Local Galois Symbols on E ×

This article studies the Albanese kernel TF (E × 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 TF (E ×E)/p to the Galois cohomology group H(F, E[p] ⊗ E[p]), for E/F ordinary, without requiring that the ptorsion 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 Ip. Non-trivial classes in the image are also constructed when E[p] is suitably unramified. A forthcoming sequel will deal with global questions.