Arithmetic Study Group: Modular Forms and Elliptic Curves over Number Fields
7 October 2014 13:00 in CM103
The celebrated connection between elliptic curves and weight 2 newforms over the rationals has a conjectural extension to general number fields. For example, over odd degree totally real fields, one knows how to associate an elliptic curve to a weight 2 newform with integer Hecke eigenvalues. Conversely, very recent work of Freitas, Hun and Siksek show that over totally real fields, most elliptic curves are modular (in fact, over real quadratic fields, "all" are modular).
Beyond totally real fields, we are at a loss at associating elliptic curves to weight 2 newforms. The best one can do
is to "search" for the elliptic curve. In joint work with X.Guitart (Essen) and M.Masdeu (Warwick), we generalize
Darmon's conjectural construction of algebraic points on elliptic curves to general number fields and then use this
conjectural construction to analytically construct the elliptic curve starting from a weight 2 newform over a general
number field, under some hypothesis. In the talk, I will start with a discussion of the first paragraph and
then will sketch our method.
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