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 \(\prod _{i\ge 1} \left( 1-\frac{1}{p^i}\right) \approx 1-\frac{1}{p}-\frac{1}{p^2}\) ${\prod }_{i\ge 1}\left(1,-,\frac{1}{p^i}\right)\approx 1-\frac{1}{p}-\frac{1}{p^2}$ . This heuristic is supported by computations which indicate that most elliptic curves (satisfying further conditions) have smooth deformation rings at a given prime \(p\ge 5\) $p\ge 5$ , and this proportion comes close to \(100\%\) $100\%$ as p gets larger.