On 2-Selmer groups of twists after quadratic extension
Adam Morgan, Ross Paterson
TL;DR
This work analyzes the $2$-Selmer groups of quadratic twists $E_d$ of an elliptic curve $E/\mathbb{Q}$ with full rational $2$-torsion, over a fixed quadratic extension $K/\mathbb{Q}$ as $d$ varies. By combining an explicit local-norm description with a Vanishing-of-an Auxiliary-Selmer result, the authors establish that for $100\%$ of twists the dimension $\dim\mathrm{Sel}^2(E_d/K)$ is governed by a local formula and follows an Erdős--Kac type distribution; they also deduce the asymptotic distribution of $\mathrm{Sha}(E_d/K)[2]$ and structural results for $E_d(K)$. The key analytical input reduces the global problem to estimating Jacobi-symbol sums, mirroring strategies in class-group analogues, and culminates in a complete description via the Weil restriction $A=\mathrm{Res}_{K/\mathbb{Q}}E$ and an isogeny framework. A complementary thin-subfamily analysis shows genuine statistical phenomena, including nontrivial $\mathrm{Gal}(K/\mathbb{Q})$-action in a positive proportion of prime twists of the congruent number curve. Overall, the paper reveals a sharp contrast between the distribution of $2$-Selmer groups over $K$ and over $\mathbb{Q}$, highlighting the influence of base-field extension on Selmer and Shafarevich–Tate statistics.
Abstract
Let $E/\mathbb{Q}$ be an elliptic curve with full rational 2-torsion. As d varies over squarefree integers, we study the behaviour of the quadratic twists $E_d$ over a fixed quadratic extension $K/\mathbb{Q}$. We prove that for 100% of twists the dimension of the 2-Selmer group over K is given by an explicit local formula, and use this to show that this dimension follows an Erdős--Kac type distribution. This is in stark contrast to the distribution of the dimension of the corresponding 2-Selmer groups over $\mathbb{Q}$, and this discrepancy allows us to determine the distribution of the 2-torsion in the Shafarevich--Tate groups of the $E_d$ over K also. As a consequence of our methods we prove that, for 100% of twists d, the action of $\operatorname{Gal}(K/\mathbb{Q})$ on the 2-Selmer group of $E_d$ over K is trivial, and the Mordell--Weil group $E_d(K)$ splits integrally as a direct sum of its invariants and anti-invariants. On the other hand, we give examples of thin families of quadratic twists in which a positive proportion of the 2-Selmer groups over K have non-trivial $\operatorname{Gal}(K/\mathbb{Q})$-action, illustrating that the previous results are genuinely statistical phenomena.