  This article is the first part of a two-part work on the Albanese kernel
T_F(E x E), for an elliptic curve E over 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 \times E)/p to the Galois cohomology group H^2(F, E[p]
(x) E[p]), for F a p-adic field and E/F ordinary, without requiring that the
p-torsion points are F-rational. A key step is to show that the image is zero
when the Galois module E[p] is non-semisimple. The forthcoming second part will
deal with global questions.
