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Faltings heights of abelian varieties with complex multiplication. (English) Zbl 1464.11059

Let \(E\) be a \(\text{CM}\) field of degree \(2d\) with maximal totally real subfield \(F\). Let \(A\) be an abelian variety over \(\mathbb{C}\) of dimension \(d\) with complex multiplication by the maximal order \(\mathcal{O}_E \subset E\) and having \(\text{CM}\) type \(\Phi \subset \text{Hom}(E; \mathbb{C})\). Colmez conjectured a formula for the Faltings height \(h^{\text{Falt}}_{(E; \Phi)}\) of \(A\) (which depends only on the pair \((E; \Phi)\) and not on \(A\) itself) in terms of the logarithmic derivatives at \(s=0\) of certain Artin \(L\)-functions, constructed in terms of the purely Galois-theoretic input \((E; \Phi)\). When \(d=1\), \(E\) is a quadratic imaginary field, Colmez’s conjecture is a form of the Chowla-Selberg formula.
In the present paper, the authors prove an averaged version of Colmez’s conjecture as follows: \[\frac{1}{2^d} \sum_{\Phi} h^{\text{Falt}}_{(E; \Phi)} =-\frac{1}{2} \cdot \frac{L^{\prime}(0, \chi)}{L(0, \chi)} -\frac{1}{4} \cdot \log \left|\frac{D_E}{D_F}\right| -\frac{d}{2} \cdot \log(2 \pi),\] where \(\chi: \mathbb{A}_{\mathbb{F}}^{\times} \rightarrow \{ \pm 1 \}\) is the quadratic Hecke character determined by the extension \(E/F\) and \(L(s; \chi)\) is the usual \(L\)-function without the local factors at Archimedean places. The sum on the left is over all \(\text{CM}\) types of \(E\), and \(D_E\) and \(D_F\) are the discriminants of \(E\) and \(F\), respectively. In order to prove the above formula, the authors calculate the arithmetic intersection multiplicities on Shimura varieties of type \(\text{GSpin}(n; 2)\), make essential use of the theory of Borcherds products and certain Green function calculations. As an application, J. Tsimerman [Ann. Math. (2) 187, No. 2, 379–390 (2018; Zbl 1415.11086)] proves that the above formula implies the André-Oort conjecture for the moduli space \(\mathcal{A}_g\) of principally polarized abelian varieties. Note that X. Yuan and S.-W. Zhang [Ann. Math. (2) 187, No. 2, 533–638 (2018; Zbl 1412.11078)] prove the averaged Colmez conjecture independently by the Gross-Zagier style results for Shimura curves over totally real fields.
Reviewer: Lei Yang (Beijing)

MSC:

11G15 Complex multiplication and moduli of abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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