Arithmetic statistics for Galois deformation rings. (English) Zbl 07887389
Summary: 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 \(\prod_{i \geq 1} \left(1-\frac{1}{p^i}\right) \approx 1-\frac{1}{p}-\frac{1}{p^2}\). 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.
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