On 2-Selmer groups of twists after quadratic extension

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 $\mathrm{Gal}(K/\mathbb{Q})$ 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 $\mathrm{Gal}(K/\mathbb{Q})$-action, illustrating that the previous results are genuinely statistical phenomena.

arXiv commentAdded additional hypothesis to the statement of Corollary 6.7. Other minor corrections following referee report

References in corpus (6)

• Ranks of twists of elliptic curves and Hilbert's Tenth Problem
• Random maximal isotropic subspaces and Selmer groups
• $2^{\infty }$-Selmer groups, $2^{\infty }$-class groups, and Goldfeld's conjecture
• On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve
• The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion
• Selmer groups of elliptic curves in degree $p$ extensions

Cited by in corpus (3)

• Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions
• The $4$-rank of class groups of $K(\sqrt{n})$
• The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$