Rank stability of elliptic curves in certain non-abelian 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 $\mathrm{Gal}(L/\mathbb{Q})\simeq \mathrm{Gal}(K/\mathbb{Q})⋉\mathbb{Z}/p^n\mathbb{Z}$, 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.

Version 2: 24 pages, accepted for publication in Mathematische Nachrichten

References in corpus (2)

• Rank growth of elliptic curves over $N$-th root extensions
• Arithmetic statistics and diophantine stability for elliptic curves

Cited by in corpus (2)

• Selmer stability for elliptic curves in Galois $\ell$-extensions
• A positive density of elliptic curves are diophantine stable in certain Galois extensions