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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Vydáno v:The Ramanujan journal Ročník 64; číslo 3; s. 685 - 708
Hlavní autoři: Ray, Anwesh, Weston, Tom
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.07.2024
Témata:
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
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Popis
Shrnutí: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