Arithmetic statistics for Galois deformation rings

Given an elliptic curve E defined over the rational numbers and a prime p at which E has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the p-torsion group E[p]. The deformations considered are subject to the flat condition at p. For a fixed elliptic curve without complex multiplication, it is shown that these deformation rings are unobstructed for all but finitely many primes. For a fixed prime p and varying elliptic curve E, we relate the problem to the question of how often p does not divide the modular degree. Heuristics due to M.Watkins based on those of Cohen and Lenstra indicate that this proportion should be Pi(i >= 1) (1 - 1/p(i)) approximate to 1 - 1/p - 1/p(2). This heuristic is supported by computations which indicate that most elliptic curves (satisfying further conditions) have smooth deformation rings at a given prime p >= 5, and this proportion comes close to 100% as p gets larger.