Faltings heights of abelian varieties with complex multiplication

Abstract

Let $M$ be the Shimura variety associated with the group of spinor similitudes of a quadratic space over $\mathbb {Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on $M$ to the central derivatives of certain $L$-functions.

As an application of this result, we prove an averaged version of Colmez’s conjecture on the Faltings heights of CM abelian varieties.

  • [AGHMP] Go to document F. Andreatta, E. Z. Goren, B. Howard, and K. Madapusi Pera, "Height pairings on orthogonal Shimura varieties," Compos. Math., vol. 153, iss. 3, pp. 474-534, 2017.
    @ARTICLE{AGHMP,
      author = {Andreatta, Fabrizio and Goren, Eyal Z. and Howard, Benjamin and Madapusi Pera, Keerthi},
      title = {Height pairings on orthogonal {S}himura varieties},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {153},
      year = {2017},
      number = {3},
      pages = {474--534},
      issn = {0010-437X},
      mrclass = {11G18 (14G35 14G40)},
      mrnumber = {3705233},
      doi = {10.1112/S0010437X1600779X},
      zblnumber = {06791573},
      }
  • [armitage] Go to document J. V. Armitage, "On a theorem of Hecke in number fields and function fields," Invent. Math., vol. 2, pp. 238-246, 1967.
    @ARTICLE{armitage,
      author = {Armitage, J. V.},
      title = {On a theorem of {H}ecke in number fields and function fields},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {2},
      year = {1967},
      pages = {238--246},
      issn = {0020-9910},
      mrclass = {10.66},
      mrnumber = {0213324},
      mrreviewer = {L. R. McCulloh},
      doi = {10.1007/BF01425516},
      zblnumber = {0143.06304},
      }
  • [BerthelotBreenMessingII] Go to document P. Berthelot, L. Breen, and W. Messing, Théorie de Dieudonné Cristalline. II, Springer-Verlag, Berlin, 1982, vol. 930.
    @BOOK{BerthelotBreenMessingII,
      author = {Berthelot, Pierre and Breen, Lawrence and Messing, William},
      title = {Théorie de {D}ieudonné Cristalline. {II}},
      series = {Lecture Notes in Math.},
      volume = {930},
      publisher = {Springer-Verlag, Berlin},
      year = {1982},
      pages = {x+261},
      isbn = {3-540-11556-0},
      mrclass = {14L05 (14F30)},
      mrnumber = {0667344},
      mrreviewer = {Lucile Bégueri},
      doi = {10.1007/BFb0093025},
      zblnumber = {0516.14015},
      }
  • [blasius:padic] Go to document D. Blasius, "A $p$-adic property of Hodge classes on abelian varieties," in Motives, Amer. Math. Soc., Providence, RI, 1994, vol. 55, pp. 293-308.
    @INCOLLECTION{blasius:padic,
      author = {Blasius, Don},
      title = {A {$p$}-adic property of {H}odge classes on abelian varieties},
      booktitle = {Motives},
      venue = {{S}eattle, {WA},
      1991},
      series = {Proc. Sympos. Pure Math.},
      volume = {55},
      pages = {293--308},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1994},
      mrclass = {14F30 (11G10 14C30 14K99)},
      mrnumber = {1265557},
      mrreviewer = {Abdellah Mokrane},
      zblnumber = {0821.14028},
      doi={10.1090/pspum/055.2/1265557},
      }
  • [Bor98] Go to document R. E. Borcherds, "Automorphic forms with singularities on Grassmannians," Invent. Math., vol. 132, iss. 3, pp. 491-562, 1998.
    @ARTICLE{Bor98,
      author = {Borcherds, Richard E.},
      title = {Automorphic forms with singularities on {G}rassmannians},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {132},
      year = {1998},
      number = {3},
      pages = {491--562},
      issn = {0020-9910},
      mrclass = {11F37 (11F22 14J28 17B67 57R57)},
      mrnumber = {1625724},
      mrreviewer = {I. Dolgachev},
      doi = {10.1007/s002220050232},
      zblnumber = {0919.11036},
      }
  • [breuil:griffiths] Go to document C. Breuil, "Représentations $p$-adiques semi-stables et transversalité de Griffiths," Math. Ann., vol. 307, iss. 2, pp. 191-224, 1997.
    @ARTICLE{breuil:griffiths,
      author = {Breuil, Christophe},
      title = {Représentations {$p$}-adiques semi-stables et transversalité de {G}riffiths},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {307},
      year = {1997},
      number = {2},
      pages = {191--224},
      issn = {0025-5831},
      mrclass = {14F30 (11F85)},
      mrnumber = {1428871},
      mrreviewer = {Abdellah Mokrane},
      doi = {10.1007/s002080050031},
      zblnumber = {0883.11049},
      }
  • [Bru] Go to document J. H. Bruinier, Borcherds Products on O(2, $l$) and Chern Classes of Heegner Divisors, Springer-Verlag, Berlin, 2002, vol. 1780.
    @BOOK{Bru,
      author = {Bruinier, Jan H.},
      title = {Borcherds Products on {O}(2, {$l$}) and {C}hern Classes of {H}eegner Divisors},
      series = {Lecture Notes in Math.},
      volume = {1780},
      publisher = {Springer-Verlag, Berlin},
      year = {2002},
      pages = {viii+152},
      isbn = {3-540-43320-1},
      mrclass = {11F55 (11F23 11F27 11G18)},
      mrnumber = {1903920},
      mrreviewer = {Rainer Schulze-Pillot},
      doi = {10.1007/b83278},
      zblnumber = {1004.11021},
      }
  • [Bru:prescribed] J. H. Bruinier, Borcherds products with prescribed divisor, 2016.
    @MISC{Bru:prescribed,
      author = {Bruinier, Jan H.},
      title = {{B}orcherds products with prescribed divisor},
      note = {preprint},
      year = {2016},
      arxiv = {1607.08713},
      zblnumber = {},
      }
  • [BF] Go to document J. H. Bruinier and J. Funke, "On two geometric theta lifts," Duke Math. J., vol. 125, iss. 1, pp. 45-90, 2004.
    @ARTICLE{BF,
      author = {Bruinier, Jan Hendrik and Funke, Jens},
      title = {On two geometric theta lifts},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {125},
      year = {2004},
      number = {1},
      pages = {45--90},
      issn = {0012-7094},
      mrclass = {11F55 (11F27 11F32)},
      mrnumber = {2097357},
      mrreviewer = {Matthew G. Boylan},
      doi = {10.1215/S0012-7094-04-12513-8},
      zblnumber = {1088.11030},
      }
  • [BKY] Go to document J. H. Bruinier, S. S. Kudla, and T. Yang, "Special values of Green functions at big CM points," Int. Math. Res. Not. IMRN, iss. 9, pp. 1917-1967, 2012.
    @ARTICLE{BKY,
      author = {Bruinier, Jan Hendrik and Kudla, Stephen S. and Yang, Tonghai},
      title = {Special values of {G}reen functions at big {CM} points},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2012},
      number = {9},
      pages = {1917--1967},
      issn = {1073-7928},
      mrclass = {11G18 (11F41)},
      mrnumber = {2920820},
      mrreviewer = {Yifeng Liu},
      doi = {10.1093/imrn/rnr095},
      zblnumber = {1281.11063},
      }
  • [BY] Go to document J. H. Bruinier and T. Yang, "Faltings heights of CM cycles and derivatives of $L$-functions," Invent. Math., vol. 177, iss. 3, pp. 631-681, 2009.
    @ARTICLE{BY,
      author = {Bruinier, Jan Hendrik and Yang, Tonghai},
      title = {Faltings heights of {CM} cycles and derivatives of {$L$}-functions},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {177},
      year = {2009},
      number = {3},
      pages = {631--681},
      issn = {0020-9910},
      mrclass = {11G18 (11F67 11G40 14G35 14G40)},
      mrnumber = {2534103},
      mrreviewer = {Robin de Jong},
      doi = {10.1007/s00222-009-0192-8},
      zblnumber = {1250.11061},
      }
  • [Colmez] Go to document P. Colmez, "Périodes des variétés abéliennes à multiplication complexe," Ann. of Math. (2), vol. 138, iss. 3, pp. 625-683, 1993.
    @ARTICLE{Colmez,
      author = {Colmez, Pierre},
      title = {Périodes des variétés abéliennes à multiplication complexe},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {138},
      year = {1993},
      number = {3},
      pages = {625--683},
      issn = {0003-486X},
      mrclass = {14K22 (11R42 14F30 14L05)},
      mrnumber = {1247996},
      mrreviewer = {Greg W. Anderson},
      doi = {10.2307/2946559},
      zblnumber = {0826.14028},
      }
  • [dejong:formal_rigid] Go to document A. J. de Jong, "Crystalline Dieudonné module theory via formal and rigid geometry," Inst. Hautes Études Sci. Publ. Math., iss. 82, pp. 5-96 (1996), 1995.
    @ARTICLE{dejong:formal_rigid,
      author = {de Jong, A. J.},
      title = {Crystalline {D}ieudonné module theory via formal and rigid geometry},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      number = {82},
      year = {1995},
      pages = {5--96 (1996)},
      issn = {0073-8301},
      mrclass = {14L05 (14F30)},
      mrnumber = {1383213},
      mrreviewer = {Pierre Berthelot},
      doi = {http://www.numdam.org/item?id=PMIHES_1995__82__5_0},
      zblnumber = {0864.14009},
      }
  • [dmos] Go to document P. Deligne, J. S. Milne, A. Ogus, and K. Shih, Hodge Cycles, Motives, and Shimura Varieties, Springer-Verlag, New York, 1982, vol. 900.
    @BOOK{dmos,
      author = {Deligne, Pierre and Milne, James S. and Ogus, Arthur and Shih, Kuang-yen},
      title = {Hodge Cycles, Motives, and {S}himura Varieties},
      series = {Lecture Notes in Math.},
      volume = {900},
      publisher = {Springer-Verlag, New York},
      year = {1982},
      pages = {ii+414},
      isbn = {3-540-11174-3},
      mrclass = {14Kxx (10D25 12A67 14A20 14F30 14K22)},
      mrnumber = {0654325},
      zblnumber = {0465.00010},
      doi = {10.1007/978-3-540-38955-2},
      }
  • [Gross1986-ia] Go to document B. H. Gross, "On canonical and quasicanonical liftings," Invent. Math., vol. 84, iss. 2, pp. 321-326, 1986.
    @ARTICLE{Gross1986-ia,
      author = {Gross, Benedict H.},
      title = {On canonical and quasicanonical liftings},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {84},
      year = {1986},
      number = {2},
      pages = {321--326},
      issn = {0020-9910},
      mrclass = {14L05 (11S31 14H25)},
      mrnumber = {0833193},
      mrreviewer = {K. Shiratani},
      doi = {10.1007/BF01388810},
      zblnumber = {0597.14044},
      }
  • [GS] Go to document H. Gillet and C. Soulé, "Arithmetic intersection theory," Inst. Hautes Études Sci. Publ. Math., iss. 72, pp. 93-174 (1991), 1990.
    @ARTICLE{GS,
      author = {Gillet, Henri and Soulé, Christophe},
      title = {Arithmetic intersection theory},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      number = {72},
      year = {1990},
      pages = {93--174 (1991)},
      issn = {0073-8301},
      mrclass = {14G40 (14C17 32C30)},
      mrnumber = {1087394},
      mrreviewer = {I. Dolgachev},
      url = {http://www.numdam.org/item?id=PMIHES_1990__72__93_0},
      zblnumber = {0741.14012},
      }
  • [Harris1985-tv] Go to document M. Harris, "Arithmetic vector bundles and automorphic forms on Shimura varieties. I," Invent. Math., vol. 82, iss. 1, pp. 151-189, 1985.
    @ARTICLE{Harris1985-tv,
      author = {Harris, Michael},
      title = {Arithmetic vector bundles and automorphic forms on {S}himura varieties. {I}},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {82},
      year = {1985},
      number = {1},
      pages = {151--189},
      issn = {0020-9910},
      mrclass = {11G18 (11F55 14K22)},
      mrnumber = {0808114},
      mrreviewer = {Thomas Zink},
      doi = {10.1007/BF01394784},
      zblnumber = {0598.14019},
      }
  • [HowUnitaryCM] Go to document B. Howard, "Complex multiplication cycles and Kudla-Rapoport divisors," Ann. of Math. (2), vol. 176, iss. 2, pp. 1097-1171, 2012.
    @ARTICLE{HowUnitaryCM,
      author = {Howard, Benjamin},
      title = {Complex multiplication cycles and {K}udla-{R}apoport divisors},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {176},
      year = {2012},
      number = {2},
      pages = {1097--1171},
      issn = {0003-486X},
      mrclass = {11G18 (14G35)},
      mrnumber = {2950771},
      mrreviewer = {Rolf Berndt},
      doi = {10.4007/annals.2012.176.2.9},
      zblnumber = {1327.14126},
      }
  • [HMP] B. Howard and K. Madapusi Pera, Arithmetic of Borcherds products, 2017.
    @MISC{HMP,
      author = {Howard, Benjamin and {Madapusi Pera},
      K.},
      title = {Arithmetic of {B}orcherds products},
      arxiv={1710.00347},
      year={2017},
      zblnumber = {},
      }
  • [Howard-Pappas] Go to document B. Howard and G. Pappas, "Rapoport-Zink spaces for spinor groups," Compos. Math., vol. 153, iss. 5, pp. 1050-1118, 2017.
    @ARTICLE{Howard-Pappas,
      author = {Howard, Benjamin and Pappas, Georgios},
      title = {Rapoport-{Z}ink spaces for spinor groups},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {153},
      year = {2017},
      number = {5},
      pages = {1050--1118},
      issn = {0010-437X},
      mrclass = {11G18 (14G35)},
      mrnumber = {3705249},
      doi = {10.1112/S0010437X17007011},
      zblnumber = {06728746},
      }
  • [HY] Go to document B. Howard and T. Yang, Intersections of Hirzebruch–Zagier Divisors and CM Cycles, Springer, Heidelberg, 2012, vol. 2041.
    @BOOK{HY,
      author = {Howard, Benjamin and Yang, Tonghai},
      title = {Intersections of {H}irzebruch--{Z}agier Divisors and {CM} Cycles},
      series = {Lecture Notes in Math.},
      volume = {2041},
      publisher = {Springer, Heidelberg},
      year = {2012},
      pages = {viii+140},
      isbn = {978-3-642-23978-6},
      mrclass = {11G15 (11F41 14G40)},
      mrnumber = {2951750},
      mrreviewer = {Rolf Berndt},
      doi = {10.1007/978-3-642-23979-3},
      zblnumber = {1238.11069},
      }
  • [Hormann] Go to document F. Hörmann, The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type, American Mathematical Society, Providence, RI, 2014, vol. 35.
    @BOOK{Hormann,
      author = {H{ö}rmann, Fritz},
      title = {The Geometric and Arithmetic Volume of {S}himura Varieties of Orthogonal Type},
      series = {CRM Monogr. Ser.},
      volume = {35},
      publisher = {American Mathematical Society, Providence, RI},
      year = {2014},
      pages = {vi+152},
      isbn = {978-1-4704-1912-7},
      mrclass = {11G18 (11F23 11F27 14G35 14G40)},
      mrnumber = {3244236},
      mrreviewer = {Mihran Papikian},
      zblnumber = {1317.11003},
      doi = {10.1090/crmm/035},
      }
  • [mp:2adic] Go to document W. Kim and K. Madapusi Pera, "2-adic integral canonical models," Forum Math. Sigma, vol. 4, p. 28, 2016.
    @ARTICLE{mp:2adic,
      author = {Kim, Wansu and Madapusi Pera, Keerthi},
      title = {2-adic integral canonical models},
      journal = {Forum Math. Sigma},
      fjournal = {Forum of Mathematics. Sigma},
      volume = {4},
      year = {2016},
      pages = {e28, 34},
      issn = {2050-5094},
      mrclass = {14G35 (11G18)},
      mrnumber = {3569319},
      mrreviewer = {Su-ion Ih},
      doi = {10.1017/fms.2016.23},
      zblnumber = {1362.11059},
      }
  • [kim:2-adic] Go to document W. Kim, "The classification of $p$-divisible groups over 2-adic discrete valuation rings," Math. Res. Lett., vol. 19, iss. 1, pp. 121-141, 2012.
    @ARTICLE{kim:2-adic,
      author = {Kim, Wansu},
      title = {The classification of {$p$}-divisible groups over 2-adic discrete valuation rings},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {19},
      year = {2012},
      number = {1},
      pages = {121--141},
      issn = {1073-2780},
      mrclass = {14L15 (11S20 14F30)},
      mrnumber = {2923180},
      mrreviewer = {Marc-Hubert Nicole},
      doi = {10.4310/MRL.2012.v19.n1.a10},
      zblnumber = {1284.14056},
      }
  • [kisin:f_crystals] Go to document M. Kisin, "Crystalline representations and $F$-crystals," in Algebraic Geometry and Number Theory, Birkhäuser Boston, Boston, MA, 2006, vol. 253, pp. 459-496.
    @INCOLLECTION{kisin:f_crystals,
      author = {Kisin, Mark},
      title = {Crystalline representations and {$F$}-crystals},
      booktitle = {Algebraic Geometry and Number Theory},
      series = {Progr. Math.},
      volume = {253},
      pages = {459--496},
      publisher = {Birkhäuser Boston, Boston, MA},
      year = {2006},
      mrclass = {11S20 (14F30)},
      mrnumber = {2263197},
      mrreviewer = {Martin C. Olsson},
      doi = {10.1007/978-0-8176-4532-8_7},
      zblnumber = {1184.11052},
      }
  • [KisinJAMS] Go to document M. Kisin, "Integral models for Shimura varieties of abelian type," J. Amer. Math. Soc., vol. 23, iss. 4, pp. 967-1012, 2010.
    @ARTICLE{KisinJAMS,
      author = {Kisin, Mark},
      title = {Integral models for {S}himura varieties of abelian type},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {23},
      year = {2010},
      number = {4},
      pages = {967--1012},
      issn = {0894-0347},
      mrclass = {11G18 (14G35)},
      mrnumber = {2669706},
      mrreviewer = {Jeffrey D. Achter},
      doi = {10.1090/S0894-0347-10-00667-3},
      zblnumber = {1280.11033},
      }
  • [KisinModp] Go to document M. Kisin, "${ Mod}\,p$ points on Shimura varieties of abelian type," J. Amer. Math. Soc., vol. 30, iss. 3, pp. 819-914, 2017.
    @ARTICLE{KisinModp,
      author = {Kisin, Mark},
      title = {{${\rm Mod}\,p$} points on {S}himura varieties of abelian type},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {30},
      year = {2017},
      number = {3},
      pages = {819--914},
      issn = {0894-0347},
      mrclass = {11G18 (11G10 14G35)},
      mrnumber = {3630089},
      doi = {10.1090/jams/867},
      zblnumber = {06699460},
      }
  • [KudlaRallis] Go to document S. S. Kudla and S. Rallis, "A regularized Siegel-Weil formula: the first term identity," Ann. of Math. (2), vol. 140, iss. 1, pp. 1-80, 1994.
    @ARTICLE{KudlaRallis,
      author = {Kudla, Stephen S. and Rallis, Stephen},
      title = {A regularized {S}iegel-{W}eil formula: the first term identity},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {140},
      year = {1994},
      number = {1},
      pages = {1--80},
      issn = {0003-486X},
      mrclass = {11F70 (11F27 22E55)},
      mrnumber = {1289491},
      mrreviewer = {Colette M\oe glin},
      doi = {0.2307/2118540},
      zblnumber = {0818.11024},
      }
  • [KRY2] Go to document S. S. Kudla, M. Rapoport, and T. Yang, "Derivatives of Eisenstein series and Faltings heights," Compos. Math., vol. 140, iss. 4, pp. 887-951, 2004.
    @ARTICLE{KRY2,
      author = {Kudla, Stephen S. and Rapoport, Michael and Yang, Tonghai},
      title = {Derivatives of {E}isenstein series and {F}altings heights},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {140},
      year = {2004},
      number = {4},
      pages = {887--951},
      issn = {0010-437X},
      mrclass = {11G18 (11F30 11F37 11G50 14G35 14G40)},
      mrnumber = {2059224},
      mrreviewer = {Jens Funke},
      doi = {10.1112/S0010437X03000459},
      zblnumber = {1088.11050},
      }
  • [KRY3] Go to document S. S. Kudla, M. Rapoport, and T. Yang, Modular Forms and Special Cycles on Shimura Curves, Princeton University Press, Princeton, NJ, 2006, vol. 161.
    @BOOK{KRY3,
      author = {Kudla, Stephen S. and Rapoport, Michael and Yang, Tonghai},
      title = {Modular Forms and Special Cycles on {S}himura Curves},
      series = {Ann. of Math. Stud.},
      volume = {161},
      publisher = {Princeton University Press, Princeton, NJ},
      year = {2006},
      pages = {x+373},
      isbn = {978-0-691-12551-0; 0-691-12551-1},
      mrclass = {11G18 (11-02 11F66 14G35 14G40)},
      mrnumber = {2220359},
      mrreviewer = {Jens Funke},
      doi = {10.1515/9781400837168},
      zblnumber = {1157.11027},
      }
  • [KuAnnals] Go to document S. S. Kudla, "Central derivatives of Eisenstein series and height pairings," Ann. of Math. (2), vol. 146, iss. 3, pp. 545-646, 1997.
    @ARTICLE{KuAnnals,
      author = {Kudla, Stephen S.},
      title = {Central derivatives of {E}isenstein series and height pairings},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {146},
      year = {1997},
      number = {3},
      pages = {545--646},
      issn = {0003-486X},
      mrclass = {11F46 (11F30 11G18 14G40)},
      mrnumber = {1491448},
      mrreviewer = {Martin L. Karel},
      doi = {10.2307/2952456},
      zblnumber = {0990.11032},
      }
  • [KuBorcherds] Go to document S. S. Kudla, "Integrals of Borcherds forms," Compositio Math., vol. 137, iss. 3, pp. 293-349, 2003.
    @ARTICLE{KuBorcherds,
      author = {Kudla, Stephen S.},
      title = {Integrals of {B}orcherds forms},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {137},
      year = {2003},
      number = {3},
      pages = {293--349},
      issn = {0010-437X},
      mrclass = {11F30 (11F55 11G18 11M36 14G35)},
      mrnumber = {1988501},
      mrreviewer = {Rainer Schulze-Pillot},
      doi = {10.1023/A:1024127100993},
      zblnumber = {1046.11027},
      }
  • [kudla:special_cycles] Go to document S. S. Kudla, "Special cycles and derivatives of Eisenstein series," in Heegner Points and Rankin $L$-Series, Cambridge Univ. Press, Cambridge, 2004, vol. 49, pp. 243-270.
    @INCOLLECTION{kudla:special_cycles,
      author = {Kudla, Stephen S.},
      title = {Special cycles and derivatives of {E}isenstein series},
      booktitle = {Heegner Points and {R}ankin {$L$}-Series},
      series = {Math. Sci. Res. Inst. Publ.},
      volume = {49},
      pages = {243--270},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2004},
      mrclass = {11G18 (11F37 11F46 11F67 14G40)},
      mrnumber = {2083214},
      mrreviewer = {Alexey A. Panchishkin},
      doi = {10.1017/CBO9780511756375.009},
      zblnumber = {1073.11042},
      }
  • [KYEis] Go to document S. S. Kudla and T. Yang, "Eisenstein series for SL(2)," Sci. China Math., vol. 53, iss. 9, pp. 2275-2316, 2010.
    @ARTICLE{KYEis,
      author = {Kudla, Stephen S. and Yang, TongHai},
      title = {Eisenstein series for {SL}(2)},
      journal = {Sci. China Math.},
      fjournal = {Science China. Mathematics},
      volume = {53},
      year = {2010},
      number = {9},
      pages = {2275--2316},
      issn = {1674-7283},
      mrclass = {11F30 (11G18 11M36)},
      mrnumber = {2718827},
      mrreviewer = {Robin de Jong},
      doi = {10.1007/s11425-010-4097-1},
      zblnumber = {1266.11071},
      }
  • [KYtheta] Go to document S. S. Kudla and T. Yang, "On the pullback of an arithmetic theta function," Manuscripta Math., vol. 140, iss. 3-4, pp. 393-440, 2013.
    @ARTICLE{KYtheta,
      author = {Kudla, Stephen S. and Yang, Tonghai},
      title = {On the pullback of an arithmetic theta function},
      journal = {Manuscripta Math.},
      fjournal = {Manuscripta Mathematica},
      volume = {140},
      year = {2013},
      number = {3-4},
      pages = {393--440},
      issn = {0025-2611},
      mrclass = {11G18 (11F27 11F30 11G15)},
      mrnumber = {3019133},
      mrreviewer = {Rolf Berndt},
      doi = {10.1007/s00229-012-0569-7},
      zblnumber = {1293.11077},
      }
  • [mp:tatek3] Go to document K. Madapusi Pera, "The Tate conjecture for K3 surfaces in odd characteristic," Invent. Math., vol. 201, pp. 625-668, 2015.
    @ARTICLE{mp:tatek3,
      author = {Madapusi Pera, Keerthi},
      title = {The {T}ate conjecture for {K3} surfaces in odd characteristic},
      journal = {Invent. Math.},
      volume = {201},
      pages = {625--668},
      year={2015},
      zblnumber = {1329.14079},
      doi = {10.1007/s00222-014-0557-5},
      mrnumber = {3370622},
     }
  • [MadapusiSpin] Go to document K. Madapusi Pera, "Integral canonical models for spin Shimura varieties," Compos. Math., vol. 152, iss. 4, pp. 769-824, 2016.
    @ARTICLE{MadapusiSpin,
      author = {Madapusi Pera, Keerthi},
      title = {Integral canonical models for spin {S}himura varieties},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {152},
      year = {2016},
      number = {4},
      pages = {769--824},
      issn = {0010-437X},
      mrclass = {11G18 (14G35)},
      mrnumber = {3484114},
      mrreviewer = {Shuichiro Takeda},
      doi = {10.1112/S0010437X1500740X},
      zblnumber = {06781978},
      }
  • [Madapusi] Go to document K. S. Madapusi Sampath, Toroidal Compactifications of Integral Models of Shimura Varieties of Hodge Type, ProQuest LLC, Ann Arbor, MI, 2011.
    @BOOK{Madapusi,
      author = {Madapusi Sampath, Keerthi Shyam},
      title = {Toroidal Compactifications of Integral Models of {S}himura Varieties of {H}odge Type},
      note = {thesis ({Ph.D.})--{T}he {U}niversity of {C}hicago},
      publisher = {ProQuest LLC, Ann Arbor, MI},
      year = {2011},
      pages = {184},
      isbn = {978-1124-71794-4},
      mrclass = {Thesis},
      mrnumber = {2898617},
      url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3460211},
      zblnumber = {},
      }
  • [MessingBT] Go to document W. Messing, The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Schemes, Springer-Verlag, New York, 1972, vol. 264.
    @BOOK{MessingBT,
      author = {Messing, William},
      title = {The Crystals Associated to {B}arsotti-{T}ate Groups: With Applications to Abelian Schemes},
      series = {Lecture Notes in Math.},
      volume = {264},
      publisher = {Springer-Verlag, New York},
      year = {1972},
      pages = {iii+190},
      mrclass = {14L05 (14B20 14D15 14F30 14K10)},
      mrnumber = {0347836},
      mrreviewer = {F. Oort},
      zblnumber = {0243.14013},
      doi = {10.1007/BFb0058301},
      }
  • [Obus] Go to document A. Obus, "On Colmez’s product formula for periods of CM-abelian varieties," Math. Ann., vol. 356, iss. 2, pp. 401-418, 2013.
    @ARTICLE{Obus,
      author = {Obus, Andrew},
      title = {On {C}olmez's product formula for periods of {CM}-abelian varieties},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {356},
      year = {2013},
      number = {2},
      pages = {401--418},
      issn = {0025-5831},
      mrclass = {14K15 (11G15 11G20 14K22)},
      mrnumber = {3048601},
      mrreviewer = {Fumio Hazama},
      doi = {10.1007/s00208-012-0855-4},
      zblnumber = {1357.11059},
      }
  • [Ogus2001-wy] A. Ogus, "Singularities of the height strata in the moduli of $K3$ surfaces," in Moduli of Abelian Varieties, Birkhäuser, Basel, 2001, vol. 195, pp. 325-343.
    @INCOLLECTION{Ogus2001-wy,
      author = {Ogus, Arthur},
      title = {Singularities of the height strata in the moduli of {$K3$} surfaces},
      booktitle = {Moduli of Abelian Varieties},
      venue = {{T}exel {I}sland, 1999},
      series = {Progr. Math.},
      volume = {195},
      pages = {325--343},
      publisher = {Birkhäuser, Basel},
      year = {2001},
      mrclass = {14J28 (14F30)},
      mrnumber = {1827026},
      mrreviewer = {Vasile Br\^\i nz\u anescu},
      zblnumber = {1022.14009},
      }
  • [rapoport_zink] Go to document M. Rapoport and T. Zink, Period Spaces for $p$-Divisible Groups, Princeton University Press, Princeton, NJ, 1996, vol. 141.
    @BOOK{rapoport_zink,
      author = {Rapoport, M. and Zink, Th.},
      title = {Period Spaces for {$p$}-Divisible Groups},
      series = {Ann. of Math. Stud.},
      volume = {141},
      publisher = {Princeton University Press, Princeton, NJ},
      year = {1996},
      pages = {xxii+324},
      isbn = {0-691-02782-X; 0-691-02781-1},
      mrclass = {14G20 (11G18 14F30 14L05 14M15 20G05 20G25)},
      mrnumber = {1393439},
      mrreviewer = {Robert E. Kottwitz},
      doi = {10.1515/9781400882601},
      zblnumber = {0873.14039},
      }
  • [ShimuraQuadratic] Go to document G. Shimura, Arithmetic of Quadratic Forms, Springer, New York, 2010.
    @BOOK{ShimuraQuadratic,
      author = {Shimura, Goro},
      title = {Arithmetic of Quadratic Forms},
      series = {Springer Monogr. Math.},
      publisher = {Springer, New York},
      year = {2010},
      pages = {xii+237},
      isbn = {978-1-4419-1731-7},
      mrclass = {11-02 (11D09 11E08 11E12 11E25 11E41)},
      mrnumber = {2665139},
      mrreviewer = {A. G. Earnest},
      doi = {10.1007/978-1-4419-1732-4},
      zblnumber = {1202.11026},
      }
  • [SouleBook] Go to document C. Soulé, Lectures on Arakelov Geometry, Cambridge University Press, Cambridge, 1992, vol. 33.
    @BOOK{SouleBook,
      author = {Soulé, C.},
      title = {Lectures on {A}rakelov Geometry},
      series = {Cambridge Stud. Adv. Math.},
      volume = {33},
      note = {with the collaboration of {D. A}bramovich, {J.-F. B}urnol and {J. K}ramer},
      publisher = {Cambridge University Press, Cambridge},
      year = {1992},
      pages = {viii+177},
      isbn = {0-521-41669-8},
      mrclass = {14G40 (11G35 14C17 58G26)},
      mrnumber = {1208731},
      mrreviewer = {Dipendra Prasad},
      doi = {10.1017/CBO9780511623950},
      zblnumber = {0812.14015},
      }
  • [Tsimerman] Go to document J. Tsimerman, "A proof of the André-Oort conjecture for $A_g$," Ann. of Math. (2), vol. 187, iss. 2, pp. 379-390, 2018.
    @article{Tsimerman,
      author = {Tsimerman, J.},
      title = {A proof of the {A}ndré-{O}ort conjecture for {$A_g$}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {187},
      year = {2018},
      number = {2},
      pages = {379--390},
      issn = {0003-486X},
      doi = {10.4007/annals.2018/187.2.2},
      zblnumber = {},
      }
  • [vologodsky:hodge] Go to document V. Vologodsky, "Hodge structure on the fundamental group and its application to $p$-adic integration," Mosc. Math. J., vol. 3, iss. 1, pp. 205-247, 2003.
    @ARTICLE{vologodsky:hodge,
      author = {Vologodsky, Vadim},
      title = {Hodge structure on the fundamental group and its application to {$p$}-adic integration},
      journal = {Mosc. Math. J.},
      fjournal = {Moscow Mathematical Journal},
      volume = {3},
      year = {2003},
      number = {1},
      pages = {205--247},
      issn = {1609-3321},
      mrclass = {14D10 (14D07)},
      mrnumber = {1996809},
      mrreviewer = {Bruno Chiarellotto},
      zblnumber = {1050.14013},
      doi = {10.1007/978-3-0348-8303-0_12},
     }
  • [Yang] Go to document T. Yang, "CM number fields and modular forms," Pure Appl. Math. Q., vol. 1, iss. 2, Special Issue: In memory of Armand Borel. Part 1, pp. 305-340, 2005.
    @ARTICLE{Yang,
      author = {Yang, Tonghai},
      title = {C{M} number fields and modular forms},
      journal = {Pure Appl. Math. Q.},
      fjournal = {Pure and Applied Mathematics Quarterly},
      volume = {1},
      year = {2005},
      number = {2, Special Issue: In memory of Armand Borel. Part 1},
      pages = {305--340},
      issn = {1558-8599},
      mrclass = {11F41 (11R80)},
      mrnumber = {2194727},
      mrreviewer = {Jens Funke},
      doi = {10.4310/PAMQ.2005.v1.n2.a5},
      zblnumber = {1146.11028},
      }
  • [Yang:colmez] Go to document T. Yang, "Arithmetic intersection on a Hilbert modular surface and the Faltings height," Asian J. Math., vol. 17, iss. 2, pp. 335-381, 2013.
    @ARTICLE{Yang:colmez,
      author = {Yang, Tonghai},
      title = {Arithmetic intersection on a {H}ilbert modular surface and the {F}altings height},
      journal = {Asian J. Math.},
      fjournal = {Asian Journal of Mathematics},
      volume = {17},
      year = {2013},
      number = {2},
      pages = {335--381},
      issn = {1093-6106},
      mrclass = {11G15 (11F41 14K22)},
      mrnumber = {3078934},
      mrreviewer = {Federica Galluzzi},
      doi = {10.4310/AJM.2013.v17.n2.a4},
      zblnumber = {1298.11056},
      }
  • [YZZ] Go to document X. Yuan, S. Zhang, and W. Zhang, The Gross-Zagier Formula on Shimura Curves, Princeton University Press, Princeton, NJ, 2013, vol. 184.
    @BOOK{YZZ,
      author = {Yuan, Xinyi and Zhang, Shou-Wu and Zhang, Wei},
      title = {The {G}ross-{Z}agier Formula on {S}himura Curves},
      series = {Ann. of Math. Stud.},
      volume = {184},
      publisher = {Princeton University Press, Princeton, NJ},
      year = {2013},
      pages = {x+256},
      isbn = {978-0-691-15592-0},
      mrclass = {11G18 (11F70 14G35)},
      mrnumber = {3237437},
      mrreviewer = {Ernest Hunter Brooks},
      zblnumber = {1272.11082},
      doi = {10.23943/princeton/9780691155913.001.0001},
     }
  • [YZ] Go to document X. Yuan and S. Zhang, "On the averaged Colmez conjecture," Ann. of Math. (2), vol. 187, iss. 2, pp. 533-638, 2018.
    @ARTICLE{YZ,
      author = {Yuan, Xinyi and Zhang, Shou-Wu},
      title = {On the averaged {C}olmez conjecture},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {187},
      year = {2018},
      number = {2},
      pages = {533--638},
      issn = {0003-486X},
      doi = {10.4007/annals.2018/187.2.4},
      zblnumber = {},
      }
  • [ZinkWindows] T. Zink, "Windows for displays of $p$-divisible groups," in Moduli of Abelian Varieties, Birkhäuser, Basel, 2001, vol. 195, pp. 491-518.
    @INCOLLECTION{ZinkWindows,
      author = {Zink, Thomas},
      title = {Windows for displays of {$p$}-divisible groups},
      booktitle = {Moduli of Abelian Varieties},
      venue = {{T}exel {I}sland, 1999},
      series = {Progr. Math.},
      volume = {195},
      pages = {491--518},
      publisher = {Birkhäuser, Basel},
      year = {2001},
      mrclass = {14L05 (14F30)},
      mrnumber = {1827031},
      mrreviewer = {Oliver Bültel},
      zblnumber = {1099.14036},
      }

Authors

Fabrizio Andreatta

Dipartimento di Matematica ``Federigo Enriques", Università di Milano,\hfill\break\indent Milano, Italia

Eyal Z. Goren

Department of Mathematics and Statistics, McGill University,\hfill\break\indent Montreal, QC, Canada

Benjamin Howard

Department of Mathematics, Boston College, Chestnut Hill, MA

Keerthi Madapusi Pera

Department of Mathematics, University of Chicago, Chicago, IL

Current address:

Department of Mathematics, Boston College,\hfill\break\indent Chestnut Hill, MA