Selmer stability for elliptic curves in Galois ℓ‐extensions

Abstract We study the behavior of Selmer groups of an elliptic curve in finite Galois extensions with prescribed Galois group. Fix a prime , a finite group with , and an elliptic curve with and surjective mod‐ Galois representation. We show that there exist infinitely many Galois extensions with Galois group for which the ‐Selmer group also vanishes. We obtain an asymptotic lower bound for the number of such fields with absolute discriminant , proving that there is an explicit constant such that The asymptotic for matches the conjectural count for all ‐extensions for which , up to a power of . This demonstrates that Selmer stability is not a rare phenomenon.