In this article we establish the effective Shafarevich conjecture for abelian varieties over ${\mathbb Q}$ of ${\text {GL}_2}$ -type. The proof combines Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties.

中文翻译：

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型阿贝尔变种的有效沙法列维奇猜想

在本文中，我们建立了有效的 Shafarevich 猜想，适用于阿贝尔变体 ${\mathbb Q}$ 的 ${\text {GL}_2}$ -类型。该证明将 Faltings 的方法与 Serre 的模块化猜想、同源估计和 Arakelov 理论的结果相结合。我们的结果为有效研究某些高维模方案上的积分点开辟了道路，例如希尔伯特模变体。