We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general setup for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley–Weil theorem for stacks.

中文翻译：

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良好的减速和循环覆盖

我们使用循环覆盖证明了数域上的簇集的有限性结果，并且在给定的有限位置集之外具有良好的减少。我们获得了加权射影面、阿贝尔簇的双重覆盖的沙法列维奇猜想的一个版本，并将超曲面的沙法列维奇猜想简化为高维超曲面的情况。这些是循环覆盖模堆栈上积分点的一般设置的特殊情况，我们的算术结果是通过堆栈的 Chevalley-Weil 定理的版本获得的。