Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

TL;DR

This paper introduces databases of elliptic curves ordered by height, analyzing their ranks and Selmer groups, revealing that average rank decreases with increasing height, aligning with recent theoretical bounds.

Contribution

It provides new databases ordered by height with computed ranks and Selmer groups, and observes a decreasing average rank phenomenon in these datasets.

Findings

Average rank decreases as height increases.

Distributions of Selmer groups match theoretical predictions.

Databases enable new insights into elliptic curve properties.

Abstract

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$ ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$ ordered by height in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon observed in these databases is that the average rank eventually decreases as height increases.