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

TL;DR

This paper explores the behavior of $S$-unit equations and elliptic or hyperelliptic curves over the cyclotomic $bZ_ell$-extensions of $bQ$, showing infinite solutions and constructing infinitely many curves with specific reduction properties.

Contribution

It demonstrates the existence of infinitely many solutions to the $S$-unit equation and constructs infinitely many elliptic or hyperelliptic curves over cyclotomic extensions with controlled reduction properties, extending classical finiteness results.

Findings

Infinitely many solutions to the $S$-unit equation over certain cyclotomic extensions.

Construction of infinitely many elliptic or hyperelliptic curves with good reduction outside specified primes.

Jacobian varieties of these curves belong to infinitely many distinct isogeny classes for some primes.

Abstract

Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$, and write $\mathcal{O}_S^\times$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$, 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 $K=\mathbb{Q}_{\infty,\ell}$ which denotes the $\mathbb{Z}_\ell$-cyclotomic extension of $\mathbb{Q}$. We show that the $S$-unit equation $\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$, has infinitely many solutions for $\ell \in \{2,3,5,7\}$, where $S$ consists only of the totally ramified prime above $\ell$. Moreover, for every prime $\ell$, we construct infinitely many elliptic or hyperelliptic curves defined over $K$ with good reduction away from $2$ and $\ell$. For certain primes $\ell$ we show that the Jacobians of these curves in fact belong to infinitely many distinct isogeny classes.