Minimal modularity lifting for GL2 over an arbitrary number field
نویسنده
چکیده
We prove a modularity lifting theorem for minimally ramified deformations of two-dimensional odd Galois representations, over an arbitrary number field. The main ingredient is a generalization of the Taylor-Wiles method in which we patch complexes rather than modules.
منابع مشابه
Potential Modularity and Applications
In our seminar we have been working towards a modularity lifting theorem. Recall that such a theorem allows one (under suitable hypotheses) to deduce the modularity of a p-adic Galois representation from that of the corresponding mod p representation. This is a wonderful theorem, but it is not immediately apparent how it can be applied: when does one know that the residual representation is mod...
متن کاملModularity Lifting beyond the Taylor–wiles Method
We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions – one must be in a Shimura variety settin...
متن کاملModularity of fibres in rigid local systems
ρ : GK −→ GL2(E), where E is either a finite field of characteristic l or a finite extension of Ql. Assume that the restrictions of ρ to the inertia groups at the primes of K above l are potentially semistable in the sense of [FM]. The representation ρ is called modular if it is associated to a Hilbert modular form on GL2(K), as is explained, for example, in [W1] and [W2]. Fontaine and Mazur [F...
متن کاملMod 2 and Mod 5 Icosahedral Representations
We shall call a simple abelian variety A/Q modular if it is isogenous (over Q) to a factor of the Jacobian of a modular curve. In this paper we shall call a representation ρ̄ : GQ→GL2(F̄l) modular if there is a modular abelian variety A/Q, a number field F/Q of degree equal to dimA, an embedding OF ↪→ End(A/Q) and a homomorphism θ : OF→F̄l such that ρ̄ is equivalent to the action of GQ on the ker θ...
متن کاملOn Two Dimensional Weight Two Odd Representations of Totally Real Fields
We say that a two dimensional p-adic Galois representation GF → GL2(Qp) of a number field F is weight two if it is de Rham with Hodge-Tate weights 0 and −1 equally distributed at each place above p; for example, the Tate module of an elliptic curve has this property. The purpose of this paper is to establish a variety of results concerning odd weight two representations of totally real fields i...
متن کامل