Unbounded average Selmer ranks of elliptic curves in torsion families
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915684297998336 |
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| author | Phillips, Tristan |
| author_facet | Phillips, Tristan |
| contents | Let $M$ and $N$ be positive integers for which the modular curve $X_1(M,MN)$ has genus $0$, and let $p$ be a prime divisor of $MN$. This article gives asymptotic lower bounds for the average size of the $p$-Selmer group of elliptic curves over a number field, with torsion subgroup $\mathbb{Z}/M\mathbb{Z} \oplus \mathbb{Z}/MN\mathbb{Z}$. In many cases, it is shown that this average is unbounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unbounded average Selmer ranks of elliptic curves in torsion families Phillips, Tristan Number Theory Primary 11G05, Secondary 11K65, 11G07, 11G18, 14G35, 11G50 Let $M$ and $N$ be positive integers for which the modular curve $X_1(M,MN)$ has genus $0$, and let $p$ be a prime divisor of $MN$. This article gives asymptotic lower bounds for the average size of the $p$-Selmer group of elliptic curves over a number field, with torsion subgroup $\mathbb{Z}/M\mathbb{Z} \oplus \mathbb{Z}/MN\mathbb{Z}$. In many cases, it is shown that this average is unbounded. |
| title | Unbounded average Selmer ranks of elliptic curves in torsion families |
| topic | Number Theory Primary 11G05, Secondary 11K65, 11G07, 11G18, 14G35, 11G50 |
| url | https://arxiv.org/abs/2512.16120 |