Arithmetic statistics for Galois deformation rings - Ray, Anwesh; Weston, Tom (The Ramanujan journal, 2024)
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 ∏ i ≥ 1 1 - 1 p i ≈ 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.