Unbounded average Selmer ranks of elliptic curves in torsion families

Tristan Phillips

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arXiv · 2512.16120

Unbounded average Selmer ranks of elliptic curves in torsion families

Tristan Phillips

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Abstract

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.

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Tristan Phillips. 2025-12-18. Unbounded average Selmer ranks of elliptic curves in torsion families. https://arxiv.org/abs/2512.16120

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