Abstract We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and A-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and A-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.

中文翻译：

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CM阿贝尔簇的周期乘积公式和函数场模拟

摘要 我们考察了 Colmez 关于复数乘法阿贝尔簇的周期的法尔廷斯高度和乘积公式的理论和猜想，以及作者开发的函数场模拟。在这个类比中，阿贝尔变体被 Drinfeld 模块和 A-motives 取代。我们还解释了阿贝尔变体、Drinfeld 模和 A 动机的必要背景，包括它们的上同调理论和比较同构以及它们的复数乘法理论。