Families of elliptic curves ordered by conductor

TL;DR

This paper investigates specific families of elliptic curves ordered by conductor, providing asymptotic counts and analyzing the average size of their 2-Selmer groups, which relates to their rank distribution.

Contribution

It determines asymptotics for two special families of elliptic curves ordered by conductor and computes the average size of their 2-Selmer groups, revealing insights into their rank distribution.

Findings

Asymptotic formulas for the families ordered by conductor

Average size of 2-Selmer groups is 3 for the first family

Average rank of these elliptic curves is bounded by 1.5

Abstract

In this article, we study the family of elliptic curves $E/\mathbb{Q}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the ratio $\Delta(E)/C(E)$ is squarefree; and second, the set of elliptic curves $E$ such that $\Delta(E)/C(E)$ is bounded by a small power $(<3/4)$ of $C(E)$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. This implies that the average rank of these elliptic curves is finite, and bounded by $1.5$.