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On the averaged Colmez conjecture. (English) Zbl 1412.11078

Ann. Math. (2) 187, No. 2, 533-638 (2018); erratum ibid. 198, No. 2, 867-878 (2023).
The article under review proves an averaged version of the Colmez conjecture for general CM fields. In the case of CM elliptic curves, it is known as the Chowla-Selberg formula and the method of this article thus gives a different proof of the Chowla-Selberg formula. F. Andreatta et al. [Ann. Math. (2) 187, No. 2, 391–531 (2018; Zbl 1464.11059)] prove the averaged Colmez conjecture independently by using integral models of high-dimensional Shimura varieties and the method of T. Yang [Am. J. Math. 132, No. 5, 1275–1309 (2010; Zbl 1206.14049); Asian J. Math. 17, No. 2, 335–382 (2013; Zbl 1298.11056)].
On the other hand, the method of the article under review is divided into two parts: the first step is the comparison of the average of Faltings heights and the height by the arithmetic Hodge bundle of the canonical integral model of a Shimura curve and the second step is the comparison of the height by the arithmetic Hodge bundle and the log derivation of the \(L\)-function of the quadratic character for the CM extension. As a result, they deduce an averaged version of the Colmez conjecture which relates the average of Faltings heights and the log derivation of the \(L\)-function of the quadratic character. In particular, the second step is a hard part of this article and their proof of this part heavily depends on the Gross-Zagier formula in [X. Yuan et al., The Gross-Zagier formula on Shimura curves. Princeton, NJ: Princeton University Press (2013; Zbl 1272.11082)]. It is needless to say that the article under review proves the notable conjecture and is a fine work.

MSC:

11G15 Complex multiplication and moduli of abelian varieties
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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