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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 $3$, andBhargava and Ho have shown that the average size of the $2$-Selmer group in thefamily of elliptic curves with a marked point is exactly $6$. In contrast tothese results, we show that the average size of the $2$-Selmer group in thefamily 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 Tamagawaratio, which measures the relative sizes of the Selmer groups associated to theisogeny and its dual. Building on previous work in which we considered theTamagawa ratio in quadratic twist families, we show that, in the family of allelliptic curves with a two-torsion point, the Tamagawa ratio is essentiallygoverned by a normal distribution with mean zero and growing variance.

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