We study the behavior of Selmer groups of an elliptic curve E/Q in finite Galois extensions with prescribed Galois group. Fix a prime 5, a finite group G with \#G = ⁿ, and an elliptic curve E/Q with Sel_ (E/Q) = 0 and surjective mod- Galois representation. We show that there exist infinitely many Galois extensions F/Q with Galois group Gal (F/Q) G for which the -Selmer group Sel_ (E/F) also vanishes. We obtain an asymptotic lower bound for the number M (G, E; X) of such fields F with absolute discriminant |F| X, proving that there is an explicit constant >0 such that M (G, E; X) X^1{^{n-1 (- 1) }} (X) ^ - 1. The asymptotic for M (G, E; X) matches the conjectural count for all G-extensions F/Q for which |F| X, up to a power of X. This demonstrates that Selmer stability is not a rare phenomenon.
Pathak et al. (Tue,) studied this question.