Selmer stability for elliptic curves in Galois $\mathrm{\ell }$-extensions

We study the behavior of Selmer groups of an elliptic curve $E\mathrm{/}\mathbb{Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime $\mathrm{\ell }\ge 5$, a finite group $G$ with $\mathrm{\#}G={\mathrm{\ell }}^n$, and an elliptic curve $E\mathrm{/}\mathbb{Q}$ with $Se{l}_{\mathrm{\ell }}(E\mathrm{/}\mathbb{Q})=0$ and surjective mod-$\mathrm{\ell }$ Galois representation. We show that there exist infinitely many Galois extensions $F\mathrm{/}\mathbb{Q}$ with Galois group $Gal(F\mathrm{/}\mathbb{Q})\simeq G$ for which the $\mathrm{\ell }$-Selmer group $Se{l}_{\mathrm{\ell }}(E\mathrm{/}F)$ also vanishes. We obtain an asymptotic lower bound for the number $M(G,E;X)$ of such fields $F$ with absolute discriminant $\mathrm{\mid }{\mathrm{\Delta }}_F\mathrm{\mid }\le X$, proving that there is an explicit constant $\delta >0$ such that $M(G,E;X)\gg X^{\frac{1}{{\mathrm{\ell }}^{n-1}(\mathrm{\ell }-1)}}(\log X{)}^{\delta -1}$. The asymptotic for $M(G,E;X)$ matches the conjectural count for all $G$-extensions $F\mathrm{/}\mathbb{Q}$ for which $\mathrm{\mid }{\mathrm{\Delta }}_F\mathrm{\mid }\le X$, up to a power of $\log X$. This demonstrates that Selmer stability is not a rare phenomenon.