Arithmetic statistics for Galois deformation rings

TL;DR

This paper studies the properties of Galois deformation rings associated with elliptic curves, showing they are mostly unobstructed for large classes of primes and relating this to the divisibility of modular degrees, supported by heuristic and computational evidence.

Contribution

It establishes that Galois deformation rings for non-CM elliptic curves are mostly unobstructed at all but finitely many primes and connects this to the divisibility properties of modular degrees.

Findings

Deformation rings are unobstructed for all but finitely many primes for fixed non-CM elliptic curves.

Most elliptic curves have smooth deformation rings at primes p ≥ 5.

The proportion of curves with smooth deformation rings approaches 100% as p increases.

Abstract

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.