The modularity of elliptic curves over all but finitely many totally real fields of degree 5

Darmon points on elliptic curves over totally real fields

Let F be a number field, let K be a quadratic extension of F , and let E be an elliptic curve over F of conductor an ideal N of F . We assume that there is a newform f of weight 2 on Γ0(N ) over F (for the notion of modular forms over number fields other than Q, see, e.g., [Byg99]) whose L-function coincides with that of E. For example, this is known to be the case if F = Q by [BCDT01] and if F...

Darmon points on elliptic curves over totally real fields

Let F be a number field and let K be a quadratic extension of F . The goal of the theory of Stark-Heegner points is to generalize the construction of Heegner points on elliptic curves over quadratic imaginary fields. If F is totally real and K is totally complex (i.e., a CM field), then under fairly general hypotheses, there is a notion of Heegner points on E defined over suitable ring class fi...

2 9 A ug 2 00 7 On the modularity of supersingular elliptic curves over certain totally real number fields

We study generalisations to totally real fields of the methods originating with Wiles and Taylor-Wiles ([32], [31]). In view of the results of Skinner-Wiles [26] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some app...

Twisted Fermat curves over totally real fields

On the modularity of elliptic curves over a composite field of some real quadratic fields