Rank stability of elliptic curves in certain non-abelian extensions

Let E/Q be an elliptic curve with rank E(Q)=0. Fix an odd prime p, a positive integer n, and a finite abelian extension K/Q. In this paper, we show that there exist infinitely many extensions L/K is Galois with Gal(L/Q)similar or equal to Gal(K/Q)Z/pnZ. 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.