Selmer groups of twists of elliptic curves over K with K-rational torsion points

Abstract

We generalize a result of Frey [Fre88] on the Selmer group of twists of elliptic curves over Q with Q-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational point. We also provide examples of elliptic curves coming from [Zyw15] that satisfy the conditions of our Theorem B.

25 pages. v3: updated statements of main theorems, added Corollary E, and made various other improvements throughout