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.

设 $K$ 为一个数域和 $S$ 一组有限的非 archimedean >位2 $S$ 的 -units 组 $K$ 。 $S$ iegel 的一个著名定理断言，带有 $S$ $𝜀$ ， $\delta \in {\mathcal{𝒪}}^{\times }$ 的单位方程 $𝜀+\delta =1$ 只有有限多的解。 $S$ hafarevich 的一个著名定理断言，椭圆曲线的同构类只有有限多的， $K$ 外部 $S$ 有很好的还原。现在，它不是数字字段，而是 it $K={\mathbb{Q}}_{\infty ,\ell }$ 表示 $\mathbb{Q}$ 的 ${\mathbb{Z}}_{\ell }$ -cyclotomic 扩展。我们表明，S 单位方程 $𝜀+\delta =1$ ，其中 $𝜀$ ，有 ， $\delta \in {\mathcal{𝒪}}^{\times }$ 对 $\ell \in \{2,3,5,7\}$ 有无限多的解，其中 S 仅由上面的完全分叉素数 $\ell$ 组成。此外，对于每个素数 $\ell$ ，我们构造了无限多的椭圆或超椭圆曲线， $K$ 这些曲线在 2 和 $\ell$ 之外以良好的还原定义。对于某些素数， $\ell$ 我们表明这些曲线的雅可比行列式实际上属于无限多个不同的等生类。