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.


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)
Full Text: DOI arXiv


[1] Anderson, Greg W., Logarithmic derivatives of {D}irichlet {\(L\)}-functions and the periods of abelian varieties, Compositio Math.. Compositio Mathematica, 45, 315-332, (1982) · Zbl 0501.14025
[2] Andr\'eatta, F.; Goren, E.; Howard, B.; Madapusi-Pera, K., Faltings heights of abelian varieties with complex multiplication, Ann. of Math., 187, 391-531, (2018) · Zbl 1464.11059
[3] Boutot, J.-F.; Carayol, H., Uniformisation {\(p\)}-adique des courbes de {S}himura: les th\'eor\`“emes de \v Cerednik et de {D}rinfel\cprime d, Ast\'”erisque. Ast\'erisque, 7-45, (1991) · Zbl 0781.14010
[4] Carayol, Henri, Sur la mauvaise r\'eduction des courbes de {S}himura, Compositio Math.. Compositio Mathematica, 59, 151-230, (1986) · Zbl 0607.14021
[5] Author’s review, Towers of algebraic curves that can be uniformized by discrete subgroups of {\({\rm PGL}\sb{2}(k\sb{w})\times E\)}, Mat. Sb. (N.S.), 99(141), 211-247, (1976) · Zbl 0369.14013
[6] Selberg, Atle; Chowla, S., On {E}pstein’s zeta-function, J. Reine Angew. Math.. Journal f\"ur die Reine und Angewandte Mathematik, 227, 86-110, (1967) · Zbl 0166.05204
[7] Colmez, Pierre, P\'eriodes des vari\'et\'es ab\'eliennes \`a multiplication complexe, Ann. of Math. (2). Annals of Mathematics. Second Series, 138, 625-683, (1993) · Zbl 0826.14028
[8] Author’s review, Travaux de {S}himura. S\'eminaire {B}ourbaki, 23\`“eme ann\'”ee (1970/71), {E}xp. {N}o. 389, Lecture Notes in Mathematics, 244, 123-165, (1971)
[9] Faltings, G., Endlichkeitss\`“atze f\'”ur abelsche {V}ariet\`“aten \'”uber {Z}ahlk\"orpern, Invent. Math.. Inventiones Mathematicae, 73, 349-366, (1983) · Zbl 0588.14026
[10] Gross, Benedict H., On canonical and quasicanonical liftings, Invent. Math.. Inventiones Mathematicae, 84, 321-326, (1986) · Zbl 0597.14044
[11] Gross, Benedict H., On the periods of abelian integrals and a formula of {C}howla and {S}elberg, Invent. Math.. Inventiones Mathematicae, 45, 193-211, (1978) · Zbl 0418.14023
[12] Gross, Benedict H.; Zagier, Don B., Heegner points and derivatives of {\(L\)}-series, Invent. Math.. Inventiones Mathematicae, 84, 225-320, (1986) · Zbl 0608.14019
[13] Illusie, Luc, D\'eformations de groupes de {B}arsotti-{T}ate (d’apr\`“es {A}. {G}rothendieck), Ast\'”erisque. Ast\'erisque, 151-198, (1985) · Zbl 1182.14050
[14] Kim, Wansu, The classification of {\(p\)}-divisible groups over 2-adic discrete valuation rings, Math. Res. Lett.. Mathematical Research Letters, 19, 121-141, (2012) · Zbl 1284.14056
[15] Kisin, Mark, Crystalline representations and {\(F\)}-crystals. Algebraic Geometry and Number Theory, Progr. Math., 253, 459-496, (2006) · Zbl 1184.11052
[16] Kisin, Mark, Integral models for {S}himura varieties of abelian type, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 23, 967-1012, (2010) · Zbl 1280.11033
[17] Lau, Eike, Relations between {D}ieudonn\'e displays and crystalline {D}ieudonn\'e theory, Algebra Number Theory. Algebra & Number Theory, 8, 2201-2262, (2014) · Zbl 1308.14046
[18] Liu, Tong, The correspondence between {B}arsotti-{T}ate groups and {K}isin modules when {\(p=2\)}, J. Th\'eor. Nombres Bordeaux. Journal de Th\'eorie des Nombres de Bordeaux, 25, 661-676, (2013) · Zbl 1327.14206
[19] Messing, William, The Crystals Associated to {B}arsotti-{T}ate Groups: with Applications to Abelian Schemes, Lect. Notes in Math., 264, iii+190 pp., (1972) · Zbl 0243.14013
[20] Obus, Andrew, On {C}olmez’s product formula for periods of {CM}-abelian varieties, Math. Ann.. Mathematische Annalen, 356, 401-418, (2013) · Zbl 1357.11059
[21] Raynaud, Michel, Sch\'emas en groupes de type {\((p,\dots, p)\)}, Bull. Soc. Math. France. Bulletin de la Soci\'et\'e Math\'ematique de France, 102, 241-280, (1974) · Zbl 0325.14020
[22] Groupes de monodromie en g\'eom\'etrie alg\'ebrique. {I}, Lecture Notes in Math., 288, viii+523 pp., (1972)
[23] Tsimerman, J., The {A}ndr\'e-{O}ort conjecture for \(A_g\), Ann. of Math., 187, 379-390, (2018) · Zbl 1415.11086
[24] Waldspurger, J.-L., Sur les valeurs de certaines fonctions {\(L\)} automorphes en leur centre de sym\'etrie, Compositio Math.. Compositio Mathematica, 54, 173-242, (1985) · Zbl 0567.10021
[25] Yang, Tonghai, An arithmetic intersection formula on {H}ilbert modular surfaces, Amer. J. Math.. American Journal of Mathematics, 132, 1275-1309, (2010) · Zbl 1206.14049
[26] Yang, Tonghai, Arithmetic intersection on a {H}ilbert modular surface and the {F}altings height, Asian J. Math.. Asian Journal of Mathematics, 17, 335-381, (2013) · Zbl 1298.11056
[27] Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei, The {G}ross-{Z}agier formula on {S}himura curves, Annals of Mathematics Studies, 184, x+256 pp., (2013) · Zbl 1272.11082
[28] Zhang, Shou-Wu, Gross-{Z}agier formula for {\({\rm GL}_2\)}, Asian J. Math.. Asian Journal of Mathematics, 5, 183-290, (2001) · Zbl 1111.11030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.