Rank stability of elliptic curves in certain non‐abelian extensions

Let E/Q$E_{/\mathbb {Q}}$ be an elliptic curve with rank E(Q)=0$E(\mathbb {Q})=0$. Fix an odd prime p$p$, a positive integer n$n$, and a finite abelian extension K/Q$K/\mathbb {Q}$ with rank E(K)=0$E(K) = 0$. In this paper, we show that there exist infinitely many extensions L/K$L/K$ such that L/Q$L/\mathbb {Q}$ is Galois with Gal(L/Q)≃Gal(K/Q)⋉Z/pnZ$\operatorname{Gal}(L/\mathbb {Q}) \simeq \operatorname{Gal}(K/\mathbb {Q}) \ltimes \mathbb {Z}/p^n\mathbb {Z}$, and rank E(L)=0$E(L)=0$. This is an extension of earlier results on rank stability of elliptic curves in cyclic extensions of prime power order to a non‐abelian setting. We also obtain an asymptotic lower bound for the number of such extensions, ordered by their absolute discriminant.