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 . Fo...

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Bibliographic Details
Published in:The Ramanujan journal Vol. 64; no. 3; pp. 685 - 708
Main Authors: Ray, Anwesh, Weston, Tom
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2024
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Galois deformation rings
11G05
Distribution questions
11F80
11R45
Galois deformations
Unobstructedness
ISSN:1382-4090, 1572-9303
Online Access:Get full text
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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 ∏ 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.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-024-00839-0