Arithmetic statistics for Galois deformation rings

Anwesh Ray, Tom Weston

Published 2021-08-30Version 1

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