What question did this study set out to answer?

This research explores the average size of p-Selmer groups in elliptic curves within specific torsion families.

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.

Read Full Paperexternally

Bookmark

View Full Paper

Cite This Study

Tristan Phillips (Thu,) studied this question.

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