Selmer stability for elliptic curves in Galois ℓ‐extensions

We study the behavior of Selmer groups of an elliptic curve E/Q$ E/\mathbb {Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime ℓ≥5$ \ell \ge 5$, a finite group G$ G$ with #G=ℓn$\#G = \ell ^n$, and an elliptic curve E/Q$ E/\mathbb {Q}$ with Selℓ(E/Q)=0$ \operatorname{Sel}_\ell (E/\mathbb {Q}) = 0$ and surjective mod‐ℓ$ \ell$ Galois representation. We show that there exist infinitely many Galois extensions F/Q$ F/\mathbb {Q}$ with Galois group Gal(F/Q)≃G$ \operatorname{Gal}(F/\mathbb {Q}) \simeq G$ for which the ℓ$ \ell$‐Selmer group Selℓ(E/F)$ \operatorname{Sel}_\ell (E/F)$ also vanishes. We obtain an asymptotic lower bound for the number M(G,E;X)$ \mathcal {M}(G, E; X)$ of such fields F$F$ with absolute discriminant |ΔF|≤X$ |\Delta _F|\le X$, proving that there is an explicit constant δ>0$\delta >0$ such that M(G,E;X)≫X1ℓn−1(ℓ−1)(logX)δ−1.$$\begin{equation*} \mathcal {M}(G, E; X) \gg X^{\frac{1}{\ell ^{n-1}(\ell - 1)}} (\log X)^{\delta - 1}. \end{equation*}$$The asymptotic for M(G,E;X)$\mathcal {M}(G, E; X)$ matches the conjectural count for all G$G$‐extensions F/Q$F/\mathbb {Q}$ for which |ΔF|≤X$|\Delta _F|\le X$, up to a power of logX$\log X$. This demonstrates that Selmer stability is not a rare phenomenon.