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Arithmetic statistics for Galois deformation rings

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

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Acknowledgements

From September 2022 to September 2023, the first author’s research was supported by the CRM-Simons fellowship. We would like to thank the referee for the helpful report.

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Authors and Affiliations

  1. Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam, Siruseri, Tamil Nadu, 603103, India

    Anwesh Ray

  2. Department of Mathematics, University of Massachusetts, Amherst, MA, USA

    Tom Weston

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  1. Anwesh Ray
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  2. Tom Weston
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Both authors contributed equally in the genesis of the ideas, and the paper was simultaneously written by both authors.

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Correspondence to Anwesh Ray.

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Ray, A., Weston, T. Arithmetic statistics for Galois deformation rings. Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00839-0

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