Andreatta, Fabrizio; Goren, Eyal; Howard, Benjamin; Madapusi Pera, Keerthi Faltings heights of abelian varieties with complex multiplication. (English) Zbl 1464.11059 Ann. Math. (2) 187, No. 2, 391-531 (2018). 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) Cited in 5 ReviewsCited in 38 Documents MSC: 11G15 Complex multiplication and moduli of abelian varieties 11G18 Arithmetic aspects of modular and Shimura varieties Keywords:abelian varieties; complex multiplication; Faltings height; Shimura varieties Citations:Zbl 1415.11086; Zbl 1412.11078 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Andreatta, Fabrizio; Goren, Eyal Z.; Howard, Benjamin; Madapusi Pera, Keerthi, Height pairings on orthogonal {S}himura varieties, Compos. Math.. Compositio Mathematica, 153, 474-534, (2017) · Zbl 1428.11114 · doi:10.1112/S0010437X1600779X [2] Armitage, J. V., On a theorem of {H}ecke in number fields and function fields, Invent. Math.. Inventiones Mathematicae, 2, 238-246, (1967) · Zbl 0143.06304 · doi:10.1007/BF01425516 [3] Berthelot, Pierre; Breen, Lawrence; Messing, William, Th\'eorie de {D}ieudonn\'e Cristalline. {II}, Lecture Notes in Math., 930, x+261 pp., (1982) · Zbl 0516.14015 · doi:10.1007/BFb0093025 [4] Blasius, Don, A {\(p\)}-adic property of {H}odge classes on abelian varieties. Motives, Proc. Sympos. Pure Math., 55, 293-308, (1994) · Zbl 0821.14028 · doi:10.1090/pspum/055.2/1265557 [5] Borcherds, Richard E., Automorphic forms with singularities on {G}rassmannians, Invent. Math.. Inventiones Mathematicae, 132, 491-562, (1998) · Zbl 0919.11036 · doi:10.1007/s002220050232 [6] Breuil, Christophe, Repr\'esentations {\(p\)}-adiques semi-stables et transversalit\'e de {G}riffiths, Math. Ann.. Mathematische Annalen, 307, 191-224, (1997) · Zbl 0883.11049 · doi:10.1007/s002080050031 [7] Bruinier, Jan H., Borcherds Products on {O}(2, {\(l\)}) and {C}hern Classes of {H}eegner Divisors, Lecture Notes in Math., 1780, viii+152 pp., (2002) · Zbl 1004.11021 · doi:10.1007/b83278 [8] Bruinier, Jan H., {B}orcherds products with prescribed divisor, (2016) · Zbl 1417.11071 [9] Bruinier, Jan Hendrik; Funke, Jens, On two geometric theta lifts, Duke Math. J.. Duke Mathematical Journal, 125, 45-90, (2004) · Zbl 1088.11030 · doi:10.1215/S0012-7094-04-12513-8 [10] Bruinier, Jan Hendrik; Kudla, Stephen S.; Yang, Tonghai, Special values of {G}reen functions at big {CM} points, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 1917-1967, (2012) · Zbl 1281.11063 · doi:10.1093/imrn/rnr095 [11] Bruinier, Jan Hendrik; Yang, Tonghai, Faltings heights of {CM} cycles and derivatives of {\(L\)}-functions, Invent. Math.. Inventiones Mathematicae, 177, 631-681, (2009) · Zbl 1250.11061 · doi:10.1007/s00222-009-0192-8 [12] 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 · doi:10.2307/2946559 [13] de Jong, A. J., Crystalline {D}ieudonn\'e module theory via formal and rigid geometry, Inst. Hautes \'Etudes Sci. Publ. Math.. Institut des Hautes \'Etudes Scientifiques. Publications Math\'ematiques, 5-96, (1995) · Zbl 0864.14009 · doi:doi = {http://www.numdam.org/item?id=PMIHES_1995__82__5_0 [14] Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen, Hodge Cycles, Motives, and {S}himura Varieties, Lecture Notes in Math., 900, ii+414 pp., (1982) · Zbl 0465.00010 · doi:10.1007/978-3-540-38955-2 [15] Gross, Benedict H., On canonical and quasicanonical liftings, Invent. Math.. Inventiones Mathematicae, 84, 321-326, (1986) · Zbl 0597.14044 · doi:10.1007/BF01388810 [16] Gillet, Henri; Soul\'e, Christophe, Arithmetic intersection theory, Inst. Hautes \'Etudes Sci. Publ. Math.. Institut des Hautes \'Etudes Scientifiques. Publications Math\'ematiques, 93-174, (1990) · Zbl 0741.14012 [17] Harris, Michael, Arithmetic vector bundles and automorphic forms on {S}himura varieties. {I}, Invent. Math.. Inventiones Mathematicae, 82, 151-189, (1985) · Zbl 0598.14019 · doi:10.1007/BF01394784 [18] Howard, Benjamin, Complex multiplication cycles and {K}udla-{R}apoport divisors, Ann. of Math. (2). Annals of Mathematics. Second Series, 176, 1097-1171, (2012) · Zbl 1327.14126 · doi:10.4007/annals.2012.176.2.9 [19] Howard, Benjamin; {Madapusi Pera}, K., Arithmetic of {B}orcherds products, (2017) [20] Howard, Benjamin; Pappas, Georgios, Rapoport-{Z}ink spaces for spinor groups, Compos. Math.. Compositio Mathematica, 153, 1050-1118, (2017) · Zbl 1431.11079 · doi:10.1112/S0010437X17007011 [21] Howard, Benjamin; Yang, Tonghai, Intersections of {H}irzebruch–{Z}agier Divisors and {CM} Cycles, Lecture Notes in Math., 2041, viii+140 pp., (2012) · Zbl 1238.11069 · doi:10.1007/978-3-642-23979-3 [22] H{\"{o}}rmann, Fritz, The Geometric and Arithmetic Volume of {S}himura Varieties of Orthogonal Type, CRM Monogr. Ser., 35, vi+152 pp., (2014) · Zbl 1317.11003 · doi:10.1090/crmm/035 [23] Kim, Wansu; Madapusi Pera, Keerthi, 2-adic integral canonical models, Forum Math. Sigma. Forum of Mathematics. Sigma, 4, 28-34, (2016) · Zbl 1362.11059 · doi:10.1017/fms.2016.23 [24] 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 · doi:10.4310/MRL.2012.v19.n1.a10 [25] Kisin, Mark, Crystalline representations and {\(F\)}-crystals. Algebraic Geometry and Number Theory, Progr. Math., 253, 459-496, (2006) · Zbl 1184.11052 · doi:10.1007/978-0-8176-4532-8_7 [26] 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 · doi:10.1090/S0894-0347-10-00667-3 [27] Kisin, Mark, {\({\rm Mod}\,p\)} points on {S}himura varieties of abelian type, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 30, 819-914, (2017) · Zbl 1384.11075 · doi:10.1090/jams/867 [28] Kudla, Stephen S.; Rallis, Stephen, A regularized {S}iegel-{W}eil formula: the first term identity, Ann. of Math. (2). Annals of Mathematics. Second Series, 140, 1-80, (1994) · Zbl 0818.11024 · doi:doi = {0.2307/2118540 [29] Kudla, Stephen S.; Rapoport, Michael; Yang, Tonghai, Derivatives of {E}isenstein series and {F}altings heights, Compos. Math.. Compositio Mathematica, 140, 887-951, (2004) · Zbl 1088.11050 · doi:10.1112/S0010437X03000459 [30] Kudla, Stephen S.; Rapoport, Michael; Yang, Tonghai, Modular Forms and Special Cycles on {S}himura Curves, Ann. of Math. Stud., 161, x+373 pp., (2006) · Zbl 1157.11027 · doi:10.1515/9781400837168 [31] Kudla, Stephen S., Central derivatives of {E}isenstein series and height pairings, Ann. of Math. (2). Annals of Mathematics. Second Series, 146, 545-646, (1997) · Zbl 0990.11032 · doi:10.2307/2952456 [32] Kudla, Stephen S., Integrals of {B}orcherds forms, Compositio Math.. Compositio Mathematica, 137, 293-349, (2003) · Zbl 1046.11027 · doi:10.1023/A:1024127100993 [33] Kudla, Stephen S., Special cycles and derivatives of {E}isenstein series. Heegner Points and {R}ankin {\(L\)}-Series, Math. Sci. Res. Inst. Publ., 49, 243-270, (2004) · Zbl 1073.11042 · doi:10.1017/CBO9780511756375.009 [34] Kudla, Stephen S.; Yang, TongHai, Eisenstein series for {SL}(2), Sci. China Math.. Science China. Mathematics, 53, 2275-2316, (2010) · Zbl 1266.11071 · doi:10.1007/s11425-010-4097-1 [35] Kudla, Stephen S.; Yang, Tonghai, On the pullback of an arithmetic theta function, Manuscripta Math.. Manuscripta Mathematica, 140, 393-440, (2013) · Zbl 1293.11077 · doi:10.1007/s00229-012-0569-7 [36] Madapusi Pera, Keerthi, The {T}ate conjecture for {K3} surfaces in odd characteristic, Invent. Math., 201, 625-668, (2015) · Zbl 1329.14079 · doi:10.1007/s00222-014-0557-5 [37] Madapusi Pera, Keerthi, Integral canonical models for spin {S}himura varieties, Compos. Math.. Compositio Mathematica, 152, 769-824, (2016) · Zbl 1391.11079 · doi:10.1112/S0010437X1500740X [38] Madapusi Sampath, Keerthi Shyam, Toroidal Compactifications of Integral Models of {S}himura Varieties of {H}odge Type, 184 pp., (2011) [39] Messing, William, The Crystals Associated to {B}arsotti-{T}ate Groups: With Applications to Abelian Schemes, Lecture Notes in Math., 264, iii+190 pp., (1972) · Zbl 0243.14013 · doi:10.1007/BFb0058301 [40] 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 · doi:10.1007/s00208-012-0855-4 [41] Ogus, Arthur, Singularities of the height strata in the moduli of {\(K3\)} surfaces. Moduli of Abelian Varieties, Progr. Math., 195, 325-343, (2001) · Zbl 1022.14009 [42] Rapoport, M.; Zink, Th., Period Spaces for {\(p\)}-Divisible Groups, Ann. of Math. Stud., 141, xxii+324 pp., (1996) · Zbl 0873.14039 · doi:10.1515/9781400882601 [43] Shimura, Goro, Arithmetic of Quadratic Forms, Springer Monogr. Math., xii+237 pp., (2010) · Zbl 1202.11026 · doi:10.1007/978-1-4419-1732-4 [44] Soul\'e, C., Lectures on {A}rakelov Geometry, Cambridge Stud. Adv. Math., 33, viii+177 pp., (1992) · Zbl 0812.14015 · doi:10.1017/CBO9780511623950 [45] Tsimerman, J., A proof of the {A}ndr\'e-{O}ort conjecture for {\(A_g\)}, Ann. of Math. (2). Annals of Mathematics. Second Series, 187, 379-390, (2018) · Zbl 1415.11086 · doi:10.4007/annals.2018/187.2.2 [46] Vologodsky, Vadim, Hodge structure on the fundamental group and its application to {\(p\)}-adic integration, Mosc. Math. J.. Moscow Mathematical Journal, 3, 205-247, (2003) · Zbl 1050.14013 [47] Yang, Tonghai, C{M} number fields and modular forms, Pure Appl. Math. Q.. Pure and Applied Mathematics Quarterly, 1, 2, Special Issue: In memory of Armand Borel. Part 1, 305-340, (2005) · Zbl 1146.11028 · doi:10.4310/PAMQ.2005.v1.n2.a5 [48] 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 · doi:10.4310/AJM.2013.v17.n2.a4 [49] Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei, The {G}ross-{Z}agier Formula on {S}himura Curves, Ann. of Math. Stud., 184, x+256 pp., (2013) · Zbl 1272.11082 [50] Yuan, Xinyi; Zhang, Shou-Wu, On the averaged {C}olmez conjecture, Ann. of Math. (2). Annals of Mathematics. Second Series, 187, 533-638, (2018) · Zbl 1412.11078 · doi:10.4007/annals.2018/187.2.4 [51] Zink, Thomas, Windows for displays of {\(p\)}-divisible groups. Moduli of Abelian Varieties, Progr. Math., 195, 491-518, (2001) · Zbl 1099.14036 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.