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Unbounded average Selmer ranks of elliptic curves in torsion families | Synapse

December 22, 2025Open Access

Unbounded average Selmer ranks of elliptic curves in torsion families

Key Points

This research explores the average size of p-Selmer groups in elliptic curves within specific torsion families.
Analyzed modular curves X1(M,MN) with genus 0 conditions.
Provided asymptotic lower bounds for the p-Selmer groups.
Investigated cases with torsion subgroups including Z/MZ and Z/MNZ.
Demonstrated that the average size of p-Selmer groups can be unbounded.
Presented findings applicable to various modular curve scenarios.

Abstract

Let M and N be positive integers for which the modular curve X₁ (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 Z/MZ Z/MNZ. In many cases, it is shown that this average is unbounded.

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Cite This Study

Tristan Phillips (Thu,) studied this question.

synapsesocial.com/papers/69488bc877063b71e748cefe https://doi.org/https://doi.org/10.48550/arxiv.2512.16120

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