Abstract. We prove that all elliptic curves defined over real quadratic fields are modular.

We prove the weight part of Serre's conjecture in generic situations for forms of U (3) which are compact at infinity and split at places dividing p as conjectured by [Her09]. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights (2, 1, 0) for K/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings. M denote the O-flat and reduced quotient of R τ,β M such that Spec R τ,β,∇ M

We prove the weight elimination direction of the Serre weight conjectures as formulated by [Her09] for forms of U (n) which are compact at infinity and split at places dividing p in generic situations. That is, we show that all modular weights for a mod p Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the "obvious" weights. Contents

We prove a level raising mod ℓ = 2 theorem for elliptic curves over Q. It generalizes theorems of Ribet and Diamond-Taylor and also explains different sign phenomena compared to odd ℓ. We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.