The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point

TL;DR

This paper demonstrates that in the family of elliptic curves with a two-torsion point, the Tamagawa ratio follows a normal distribution with increasing variance, and the average 2-Selmer group size is unbounded.

Contribution

It reveals that the Tamagawa ratio in this family is normally distributed with growing variance, contrasting with previous bounded average sizes of 2-Selmer groups.

Findings

Tamagawa ratio follows a normal distribution with mean zero

Variance of the Tamagawa ratio grows without bound

Average size of 2-Selmer group is unbounded

Abstract

In recent work, Bhargava and Shankar have shown that the average size of the $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ is exactly $3$, and Bhargava and Ho have shown that the average size of the $2$-Selmer group in the family of elliptic curves with a marked point is exactly $6$. In contrast to these results, we show that the average size of the $2$-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous curves is the Tamagawa ratio, which measures the relative sizes of the Selmer groups associated to the isogeny and its dual. Building on previous work in which we considered the Tamagawa ratio in quadratic twist families, we show that, in the family of all elliptic curves with a two-torsion point, the Tamagawa ratio is essentially governed by a normal distribution with mean zero and growing variance.