Rank stability of elliptic curves in certain non-abelian extensions

Abstract

Let $E_{/Q}$ be an elliptic curve with rank $E\left(Q\right)=0$. Fix an odd prime $p$, a positive integer $n$, and a finite abelian extension $K/Q$ with rank $E\left(K\right)=0$. In this paper, we show that there exist infinitely many extensions $L/K$ such that $L/Q$ is Galois with $Gal(L/Q)\simeq Gal(K/Q)⋉Z/p^{n}Z$, and rank $E\left(L\right)=0$. 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.