Ranks, $2$-Selmer groups, and Tamagawa numbers of elliptic curves with $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$-torsion

TL;DR

This paper studies a specific family of elliptic curves with a particular torsion subgroup, computing ranks and Selmer groups for many examples, and shows that certain arithmetic averages grow without bound under standard conjectures.

Contribution

It provides the first large-scale computational analysis of elliptic curves with rac{2}{rac{8}{rac{Z}{Z}}} torsion, demonstrating unbounded growth of averages under GRH and BSD.

Findings

Average rank computed for 92% of 202,461 curves.

Average size of 2-Selmer groups tends to infinity.

Average Tamagawa product tends to infinity.

Abstract

In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over $\mathbb{Q}$ whose rational torsion subgroup is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. Conditional on GRH and BSD, we compute the rank of $92\%$ of the $202461$ curves with parameter height less than $10^3$. We also compute the size of the $2$-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.