Deformations Problems Attached to Modular

be the semisimple Galois representation associated to f by Deligne and Serre; here S is any finite set of primes containing all primes dividing N and GQ,S∪{l} is the Galois group of the maximal extension of Q unramified outside S ∪{l}. The representation ρ̄f,λ is absolutely irreducible for almost all λ; for such λ let R S f,λ denote the universal deformation ring parameterizing lifts of ρ̄f,λ (up to strict equivalence) to two dimensional representations of GQ,S∪{l} over noetherian local rings with residue field kλ (see Section 1.1 for a precise definition). Using recent work of Diamond, Flach, and Guo [5] on the Bloch–Kato conjectures for adjoint motives of modular forms we prove the following theorem.