Twist hypercubes and the distribution of $2$-Selmer ranks of elliptic curves

Abstract: The Poonen–Rains heuristics conjecture an explicit distribution for the $2$-Selmer ranks of elliptic curves over $\mathbb{Q}$. Their conjecture predicts in particular that every nonnegative integer should occur as the $2$-Selmer rank of a positive proportion of curves. This qualitative prediction has remained entirely open: prior to this work, not a single value of $r$ was known to occur with positive proportion. We prove this prediction for every $r$.

Our method organizes quadratic twists of elliptic curves into hypercubes whose $2$-Selmer ranks are tightly constrained by Poitou–Tate duality. We classify all valid rank patterns by simple graphs and use this classification to obtain the first two-sided bounds on rank densities in congruence families. In a complementary direction, we show that the $2$-Selmer rank evolves as a birth–death chain across the hypercube, and prove that this chain converges to the Poonen–Rains distribution. Analogous results hold for Jacobians of hyperelliptic curves of any genus.

This is joint work with Manjul Bhargava, Wei Ho, Ari Shnidman, and Alexander Smith.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar