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 $y^2 = Ax^4 + Bx^2 z^2 + Cz^4$ 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.