On 2-Selmer groups of twists after quadratic extension

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 Erd\H{o}s--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.