The Chow group of zero-cycles is a generalization to higher dimensions of the Picard group of a curve. For a smooth projective variety X over a field k, this group provides a fundamental geometric invariant, but unlike the case of curves very little is known about its structure, especially when k is a field of arithmetic interest.  In the mid 90's Colliot-Thélène formulated a conjecture about zero-cycles over p-adic fields.  A weaker form of this conjecture has been  established, but the general conjecture is only known for very limited classes of varieties. In this talk  I will present some joint work with Isabel Leal, where we prove this conjecture for a large family of products of elliptic curves. Our method often allows us to obtain very sharp results about the structure of the group of zero-cycles on such products and also give us some promising  global-to-local information.