Year: 2026
Type: article
Abstract: Let D ≠ 1 D\neq 1 be a fixed squarefree integer. For elliptic curves E / Q E/\mathbb {Q} , writing E D E_{D} for the quadratic twist by D D , we consider the question of how often E ( Q ) E(\mathbb {Q}) and E D ( Q ) E_{D}(\mathbb {Q}) generate E ( Q ( D ) ) E(\mathbb {Q}(\sqrt {D})) . We bound the proportion of E E , ordered by height, for which this is not the case, showing that it is very small for typical D D . The central theorem is concerned with intersections of 2 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.
DOI: 10.1090/tran/9608
Author: Ross Paterson
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7125583362