On average sizes of Selmer groups and ranks in families of elliptic curves having marked points

Abstract

We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average ranks of the elliptic curves in all of these families are bounded. Our proofs are uniform and make use of parametrizations involving various forms of $2\times 2\times 2\times 2$ and $3\times 3\times 3$ matrices that we studied in a previous paper. We also deduce that $100\%$ of genus one curves of the form $y^2=A{x}^4+B{x}^2{z}^2+C{z}^4$ with $A,B,C\in \mathbb{Z}$, when ordered by $\max \{|B{|}^2,|AC|\}$, fail the Hasse principle. Other forthcoming applications include proofs that a positive proportion of integers are (respectively, are not) the sum of two rational cubes, and a positive proportion of genus one curves in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ over $\mathbb{Q}$ fail the Hasse principle.

v2: several typos corrected, comments still welcome!