Number Theory Seminar

Speaker: Nuno Hultberg

Title: Mordell-Weil beyond number fields

Abstract: On an elliptic curve E over a number field K there are two ways to define a notion of height. There is the intrinsic notion of the Neron-Tate height compatible with the group structure and the more explicit notion the Weil height depending on an embedding into projective space. The two notions of height differ only by a bounded amount. The Mordell-Weil theorem stating that E(K) is finitely generated perfectly illustrates the power of this fact. Trying to apply the strategy to fields L lying inside the field of algebraic numbers leads to interesting subtleties. While we may still obtain information on the structure of E(L) using the Neron-Tate height, it is more difficult to use properties of L to deduce facts on the Neron-Tate height, but sometimes possible.