Quadratic Twists as Random Variables
Ross Paterson
TL;DR
The paper analyzes when the quadratic twist E_D and the original curve E over Q generate the K-rational points in K=Q(√D) by studying intersections of 2-Selmer groups Sel_2(E/Q) and Sel_2(E_D/Q). It reinterprets these intersections via corestriction Selmer structures and encodes Selmer elements through binary quartic forms, enabling average-size calculations through Bhargava–Shankar’s orbit counting framework. The main result expresses the average intersection size as a product of local densities and yields bounds showing nontrivial intersections occur for a positive but typically small proportion, decaying as (23/24)^{ω(D)}. A heuristic based on intersections of maximal isotropic subspaces, refined by genus theory, explains the observed behavior and predicts varied outcomes across families, with supporting evidence from quadratic-twist families and all elliptic curves. The combination of local-global Selmer analysis, the binary quartic correspondence, and Selmer-bundle mass formulas provides a cohesive mechanism for predicting intersections in broad elliptic-curve families and related twist families.
Abstract
Let $D\neq 1$ be a fixed squarefree integer. For elliptic curves $E/\mathbb{Q}$, writing $E_D$ for the quadratic twist by $D$, we consider the question of how often $E(\mathbb{Q})$ and $E_D(\mathbb{Q})$ generate $E(\mathbb{Q}(\sqrt{D}))$. We bound the proportion of $E/\mathbb{Q}$, ordered by height, for which this is not the case, showing that it is very small for typical $D$. The central theorem is concerned with intersections of 2-Selmer groups of quadratic twists. We establish their average size in terms of a product of local densities. We additionally propose a heuristic model for these intersections, which explains our result and similar results in the literature. This heuristic predicts further results in other families.