Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups
Alex Bartel, Adam Morgan
TL;DR
This work studies the distribution of $2$-Selmer groups for abelian varieties over $b Q$ along zero-density quadratic twist subfamilies defined by Frobenian constraints. By developing a local-to-global framework based on $2$-descents, local Tate pairings, and a detailed combinatorics of maximal isotropic subspaces, the authors compute the $r$-th moments of $ ext{Sel}_2(E_d/b Q)$ and prove a precise injection-moment distribution, parameterized by $n_b= ext{dim}\nobreak \\mathcal S_b$ and a parity bit $m_b$, with the universal distribution $\\alpha(r)$. These results yield concrete consequences for Galois module structures of Mordell–Weil groups over quadratic extensions, and they imply positive-density outcomes toward the Hasse principle for genus 1 twists and for Kummer varieties, conditional on finiteness of relevant Sha groups. The work also provides explicit probabilistic formulas and rational constants (e.g., $eta_n$ and the distribution $\\\alpha$) linking Selmer ranks to random-matrix-type models, highlighting the deep connections between arithmetic statistics and linear-algebraic encodings of descent data. Altogether, the paper advances the understanding of Selmer-rank distributions in sparse twist families and demonstrates tangible arithmetic consequences for rational points on genus 1 curves and Kummer varieties.
Abstract
We address several seemingly disparate problems in arithmetic geometry: the statistical behaviour of the Galois module structure of Mordell--Weil groups of a fixed elliptic curve over varying quadratic extensions; the frequency of failure of the Hasse principle in quadratic twist families of genus $1$ hyperelliptic curves; and the Hasse principle for Kummer varieties. The common technical ingredient for all of these is a result on the distribution of $2$-Selmer ranks in certain sparse families of quadratic twists of a given abelian variety.