Unbounded average Selmer ranks of elliptic curves in torsion families

TL;DR

This paper establishes that the average size of the p-Selmer groups of elliptic curves with specific torsion structures over number fields can grow without bound, providing new insights into the distribution of Selmer ranks.

Contribution

It provides asymptotic lower bounds for the average p-Selmer group sizes in families of elliptic curves with certain torsion subgroups, showing unbounded growth in many cases.

Findings

Average p-Selmer group size is unbounded in many torsion families.

Asymptotic lower bounds are derived for these averages.

Results depend on modular curve genus conditions.

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.