Arithmetic statistics for Galois deformation rings
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917706212573184 |
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| author | Ray, Anwesh Weston, Tom |
| author_facet | Ray, Anwesh Weston, Tom |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_13480 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Arithmetic statistics for Galois deformation rings Ray, Anwesh Weston, Tom Number Theory 11G05, 11F80 (primary), 11F11, 11F33 (secondary) 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. |
| title | Arithmetic statistics for Galois deformation rings |
| topic | Number Theory 11G05, 11F80 (primary), 11F11, 11F33 (secondary) |
| url | https://arxiv.org/abs/2108.13480 |