2-Selmer groups of twists of elliptic curves over quadratic field extensions

Abstract: Let E/Q be an elliptic curve with E[2] ⊂ E(Q). As d varies over squarefree integers, we learn the behaviour of the quadratic twists E_d over some fixed quadratic number field. We show that for 100% twists the dimension of the 2-Selmer group over K is given by an local formula and we use this to prove that this dimension follows some type of distribution. Consequently, we prove that, for 100% of twists d, the action of Gal(K/Q) on 2-Selmer group of E_d over K is trivial. At the end, we construct an example of thin families of quadratic twists in which a proportion of the 2-Selmer groups over K have non-trivial Gal(K/Q)-action, which shows that previous results are just statistical phenomena