Rank stability of elliptic curves in certain non‐abelian extensions

Siddhi Pathak and Anwesh Ray

Mathematische Nachrichten, 2025, vol. 298, issue 2, 730-753

Abstract: 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.

Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202400357

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:298:y:2025:i:2:p:730-753

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().