Rank stability of elliptic curves in certain non-abelian extensions
Siddhi Pathak, Anwesh Ray
TL;DR
This work proves rank stability for elliptic curves in certain non-abelian, metabelian extensions of $\mathbb{Q}$. By combining local Selmer-vanishing criteria, a Selmer-class parameterization of $(\mathcal{G},K)$-extensions, Wiles’ formula, and a Delange-type density argument, the authors construct infinitely many extensions $L/K/\mathbb{Q}$ with $\mathrm{Gal}(L/\mathbb{Q}) \cong B \ltimes \mathbb{Z}/p^n\mathbb{Z}$ such that $\mathrm{Sel}_p(E/L)=0$, hence $\operatorname{rank} E(L)=0$. The results extend prior rank-stability findings from cyclic central extensions to non-abelian settings and provide explicit asymptotic lower bounds for the count of such extensions by discriminant. They also establish a density-theoretic perspective consistent with Malle–Bhargava predictions for the growth of these extensions. Overall, the paper connects Selmer-theoretic criteria, Galois-module techniques, and analytic counting to demonstrate stable rank phenomena in a broad class of solvable extensions.
Abstract
Let $E_{/\mathbb{Q}}$ be an elliptic curve with rank $E(\mathbb{Q})=0$. Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K/\mathbb{Q}$ with rank $E(K) = 0$. In this paper, we show that there exist infinitely many extensions $L/K$ such that $L/\mathbb{Q}$ is Galois with $\operatorname{Gal}(L/\mathbb{Q}) \simeq \operatorname{Gal}(K/\mathbb{Q}) \ltimes \mathbb{Z}/p^n\mathbb{Z}$, and rank $E(L)=0$. 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.