Colmez [Périodes des variétés abéliennes a multiplication complexe,Ann. of Math. (2)138(3) (1993), 625–683; available athttp://www.math.jussieu.fr/∼colmez] conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at$s=0$of certain Artin$L$-functions. In a series of articles we investigate the analog of Colmez’s theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher-dimensional generalizations, so-called$A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM$A$-motive at all finite places in terms of Artin$L$-series. The latter is achieved by investigating the local shtukas associated with the$A$-motive.

中文翻译：

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具有复数乘法的 Drinfeld 模块和本地 SHUKAS 的时期

Colmez [Périodes des variétés abéliennes 一个乘法复合体，安。数学。(2)138(3) (1993), 625–683; 可在http://www.math.jussieu.fr/∼colmez] 用复数乘法猜想数域上的阿贝尔簇周期的乘积公式，并在某些情况下证明了这一点。他的猜想等价于 CM 阿贝尔簇的 Faltings 高度的对数导数公式$s=0$某些Artin的$L$-职能。在一系列文章中，我们研究了 Colmez 理论在函数域算术中的类比。那里的阿贝尔簇被 Drinfeld 模块和它们的高维概括所取代，所谓的$澳元-动机。在本文中，我们证明了 Carlitz 模块的乘积公式，并计算了 CM 周期的估值$澳元- 就 Artin 而言，所有有限地方的动机$L$-系列。后者是通过调查与$澳元-动机。