Curves with few bad primes over cyclotomic ℤℓ-extensions

Siksek, Samir; Visser, Robin

doi:10.2140/ant.2025.19.113

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Curves with few bad primes over cyclotomic $\mathbb{Z}_\ell$-extensions

Samir Siksek and Robin Visser

Vol. 19 (2025), No. 1, 113–141

DOI: 10.2140/ant.2025.19.113

Abstract

Let $K$ be a number field, and $S$ a finite set of nonarchimedean places of $K$ , and write $𝒪^{\times}$ for the group of $S$ -units of $K$ . A famous theorem of Siegel asserts that the $S$ -unit equation $𝜀 + δ = 1$ , with $𝜀$ , $δ \in 𝒪^{\times}$ , has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$ . Now instead of a number field, let $K = ℚ_{\infty, ℓ}$ which denotes the $ℤ_{ℓ}$ -cyclotomic extension of $ℚ$ . We show that the $S$ -unit equation $𝜀 + δ = 1$ , with $𝜀$ , $δ \in 𝒪^{\times}$ , has infinitely many solutions for $ℓ \in {2, 3, 5, 7}$ , where $S$ consists only of the totally ramified prime above $ℓ$ . Moreover, for every prime $ℓ$ , we construct infinitely many elliptic or hyperelliptic curves defined over $K$ with good reduction away from $2$ and $ℓ$ . For certain primes $ℓ$ we show that the Jacobians of these curves in fact belong to infinitely many distinct isogeny classes.

Keywords

Shafarevich conjecture, Abelian varieties, cyclic fields, cyclotomic fields, integral points

Mathematical Subject Classification

Primary: 11G10

Secondary: 11G05

Milestones

Received: 5 February 2023

Revised: 29 November 2023

Accepted: 13 February 2024

Published: 4 December 2024

Authors

Samir Siksek
Mathematics Institute University of Warwick Coventry United Kingdom

Robin Visser
Mathematics Institute University of Warwick Coventry United Kingdom

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Vol. 19, No. 1, 2025