Unobstructed modular deformation problems

Abstract

Let f be a newform of weight at least 3 with Fourier coefficients in a number field K. We show that the universal deformation ring of the mod lambda Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes lambda of K. We give an explicit bound on such lambda for the six known cusp forms of level 1, trivial character, and rational Fourier coefficients. We also prove a slightly weaker result for weight 2.