Curves with few bad primes over cyclotomic ℤℓ-extensions

Abstract

Let $K$ be a number field, and $S$ a finite set of nonarchimedean places of $K$, and write ${\mathcal{𝒪}}^{\times }$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $𝜀+\delta =1$, with $𝜀$, $\delta \in {\mathcal{𝒪}}^{\times }$, 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 $K={\mathbb{Q}}_{\infty ,\ell }$ which denotes the ${\mathbb{Z}}_{\ell }$-cyclotomic extension of $\mathbb{Q}$. We show that the $S$-unit equation $𝜀+\delta =1$, with $𝜀$, $\delta \in {\mathcal{𝒪}}^{\times }$, 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.