Complex multiplication cycles and Kudla-Rapoport divisors

Abstract

We study the intersections of special cycles on a unitary Shimura variety of signature $(n-1,1)$ and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla and suggest a Gross-Zagier theorem for unitary Shimura varieties.

  • [AMRT] Go to document A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth Compactifications of Locally Symmetric Varieties, Second ed., Cambridge: Cambridge Univ. Press, 2010.
    @book {AMRT, MRKEY = {2590897},
      AUTHOR = {Ash, Avner and Mumford, David and Rapoport, Michael and Tai, Yung-sheng},
      TITLE = {Smooth Compactifications of Locally Symmetric Varieties},
      SERIES = {Cambridge Math. Library},
      EDITION = {Second},
      NOTE = {with the collaboration of Peter Scholze},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2010},
      PAGES = {x+230},
      ISBN = {978-0-521-73955-9},
      MRCLASS = {14M27 (32J05 32M15)},
      MRNUMBER = {2590897},
      DOI = {10.1017/CBO9780511674693},
      ZBLNUMBER = {1209.14001},
      }
  • [BBM] Go to document P. Berthelot, L. Breen, and W. Messing, Théorie de Dieudonné Cristalline. II, New York: Springer-Verlag, 1982, vol. 930.
    @book {BBM, MRKEY = {0667344},
      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},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {x+261},
      ISBN = {3-540-11556-0},
      MRCLASS = {14L05 (14F30)},
      MRNUMBER = {0667344},
      MRREVIEWER = {Lucile B{é}gueri},
      ZBLNUMBER = {0516.14015},
      DOI = {10.1007/BFb0093025},
     }
  • [bruinier99] Go to document J. H. Bruinier, "Borcherds products and Chern classes of Hirzebruch-Zagier divisors," Invent. Math., vol. 138, iss. 1, pp. 51-83, 1999.
    @article {bruinier99, MRKEY = {1714336},
      AUTHOR = {Bruinier, Jan Hendrik},
      TITLE = {Borcherds products and {C}hern classes of {H}irzebruch-{Z}agier divisors},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {138},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {51--83},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F41 (11F55)},
      MRNUMBER = {1714336},
      MRREVIEWER = {Rainer Schulze-Pillot},
      DOI = {10.1007/s002220050341},
      ZBLNUMBER = {1011.11027},
      }
  • [bruinier-burgos-kuhn] Go to document J. H. Bruinier, J. I. Burgos~Gil, and U. Kühn, "Borcherds products and arithmetic intersection theory on Hilbert modular surfaces," Duke Math. J., vol. 139, iss. 1, pp. 1-88, 2007.
    @article {bruinier-burgos-kuhn, MRKEY = {2322676},
      AUTHOR = {Bruinier, Jan Hendrik and Burgos~{G}il, Jos{é} I. and K{ü}hn, Ulf},
      TITLE = {Borcherds products and arithmetic intersection theory on {H}ilbert modular surfaces},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {139},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {1--88},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11G18 (11F41 14C17 14G40)},
      MRNUMBER = {2322676},
      MRREVIEWER = {Fabrizio Andreatta},
      DOI = {10.1215/S0012-7094-07-13911-5},
      ZBLNUMBER = {1208.11077},
      }
  • [BKK] Go to document J. I. Burgos~Gil, J. Kramer, and U. Kühn, "Cohomological arithmetic Chow rings," J. Inst. Math. Jussieu, vol. 6, pp. 1-172, 2007.
    @article{BKK,
      author={Burgos~{G}il, J. I. and Kramer, J. and K{ü}hn, U.},
      TITLE={Cohomological arithmetic {C}how rings},
      JOURNAL={J. Inst. Math. Jussieu},
      VOLUME={6},
      PAGES={1--172},
      YEAR={2007},
      MRNUMBER = {2285241},
      ZBLNUMBER = {1115.14013},
      DOI = {10.1017/S1474748007000011},
     }
  • [cogdell85] Go to document J. W. Cogdell, "Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus," J. Reine Angew. Math., vol. 357, pp. 115-137, 1985.
    @article {cogdell85, MRKEY = {0783537},
      AUTHOR = {Cogdell, J. W.},
      TITLE = {Arithmetic cycles on {P}icard modular surfaces and modular forms of {N}ebentypus},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {357},
      YEAR = {1985},
      PAGES = {115--137},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11F41 (14J20)},
      MRNUMBER = {0783537},
      MRREVIEWER = {E. Gottschling},
      DOI = {10.1515/crll.1985.357.115},
      ZBLNUMBER = {0547.10028},
      }
  • [conrad04] Go to document B. Conrad, "Gross-Zagier revisited," in Heegner Points and Rankin $L$-Series, Cambridge: Cambridge Univ. Press, 2004, vol. 49, pp. 67-163.
    @incollection {conrad04, MRKEY = {2083211},
      AUTHOR = {Conrad, Brian},
      TITLE = {Gross-{Z}agier revisited},
      BOOKTITLE = {Heegner Points and {R}ankin {$L$}-Series},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {49},
      PAGES = {67--163},
      NOTE = {with an appendix by W. R. Mann},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2004},
      MRCLASS = {11G18 (11F66 11F67 11G40)},
      MRNUMBER = {2083211},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.1017/CBO9780511756375.006},
      ZBLNUMBER = {1072.11040},
      }
  • [conrad:shimura-taniyama] Go to document B. Conrad, Shimura-Taniyama formula, 2005.
    @misc{conrad:shimura-taniyama,
      author = {Conrad, Brian},
      TITLE = {Shimura-{T}aniyama formula},
      NOTE={in {\em Notes on complex multiplication: 2004-05 VIGRE number theory working group}},
      URL={http://math.stanford.edu/~conrad},
      YEAR={2005},
     }
  • [gillet84] Go to document H. Gillet, "Intersection theory on algebraic stacks and $Q$-varieties," in Proceedings of the Luminy Conference on Algebraic $K$-Theory, 1984, pp. 193-240.
    @inproceedings {gillet84, MRKEY = {0772058},
      AUTHOR = {Gillet, Henri},
      TITLE = {Intersection theory on algebraic stacks and {$Q$}-varieties},
      BOOKTITLE = {Proceedings of the {L}uminy Conference on Algebraic {$K$}-Theory},
      VENUE={{L}uminy, 1983},
      SERIES = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {34},
      YEAR = {1984},
      NUMBER = {2-3},
      PAGES = {193--240},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {14C35 (14C17 18F25 19E20)},
      MRNUMBER = {0772058},
      MRREVIEWER = {Daniel R. Grayson},
      DOI = {10.1016/0022-4049(84)90036-7},
      ZBLNUMBER = {0607.14004},
      }
  • [gillet09] H. Gillet, "Arithmetic intersection theory on Deligne-Mumford stacks," in Motives and Algebraic Cycles, Providence, RI: Amer. Math. Soc., 2009, vol. 56, pp. 93-109.
    @incollection {gillet09, MRKEY = {2562454},
      AUTHOR = {Gillet, Henri},
      TITLE = {Arithmetic intersection theory on {D}eligne-{M}umford stacks},
      BOOKTITLE = {Motives and Algebraic Cycles},
      SERIES = {Fields Inst. Commun.},
      VOLUME = {56},
      PAGES = {93--109},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      MRCLASS = {14G40 (14A20 14C17 19E08)},
      MRNUMBER = {2562454},
      MRREVIEWER = {Harry Tamvakis},
      ZBLNUMBER = {1196.14022},
      }
  • [gillet-soule90] 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 {gillet-soule90, MRKEY = {1087394},
      AUTHOR = {Gillet, Henri and Soul{é},
      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},
      CODEN = {PMIHA6},
      MRCLASS = {14G40 (14C17 32C30)},
      MRNUMBER = {1087394},
      MRREVIEWER = {I. Dolgachev},
      URL = {http://www.numdam.org/item?id=PMIHES_1990__72__93_0},
      ZBLNUMBER = {0741.14012},
      }
  • [gross86] Go to document B. H. Gross, "On canonical and quasi-canonical liftings," Invent. Math., vol. 84, iss. 2, pp. 321-326, 1986.
    @article {gross86, MRKEY = {0833193},
      AUTHOR = {Gross, Benedict H.},
      TITLE = {On canonical and quasi-canonical liftings},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {84},
      YEAR = {1986},
      NUMBER = {2},
      PAGES = {321--326},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14L05 (11S31 14H25)},
      MRNUMBER = {0833193},
      MRREVIEWER = {K. Shiratani},
      DOI = {10.1007/BF01388810},
      ZBLNUMBER = {0597.14044},
     }
  • [GKZ] Go to document B. H. Gross, W. Kohnen, and D. Zagier, "Heegner points and derivatives of $L$-series. II," Math. Ann., vol. 278, iss. 1-4, pp. 497-562, 1987.
    @article {GKZ, MRKEY = {0909238},
      AUTHOR = {Gross, Benedict H. and Kohnen, W. and Zagier, D.},
      TITLE = {Heegner points and derivatives of {$L$}-series. {II}},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {278},
      YEAR = {1987},
      NUMBER = {1-4},
      PAGES = {497--562},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11G40 (11F30 11G05 11R29 14G10)},
      MRNUMBER = {0909238},
      MRREVIEWER = {Lawrence Washington},
      DOI = {10.1007/BF01458081},
      ZBLNUMBER = {0641.14013},
      }
  • [gross-zagier85] Go to document B. H. Gross and D. B. Zagier, "On singular moduli," J. Reine Angew. Math., vol. 355, pp. 191-220, 1985.
    @article {gross-zagier85, MRKEY = {0772491},
      AUTHOR = {Gross, Benedict H. and Zagier, Don B.},
      TITLE = {On singular moduli},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {355},
      YEAR = {1985},
      PAGES = {191--220},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11F03 (11G05 14K15)},
      MRNUMBER = {0772491},
      MRREVIEWER = {R. W. K. Odoni},
      DOI = {10.1515/crll.1985.355.191},
      ZBLNUMBER = {0545.10015},
      }
  • [gross-zagier] Go to document B. H. Gross and D. B. Zagier, "Heegner points and derivatives of $L$-series," Invent. Math., vol. 84, iss. 2, pp. 225-320, 1986.
    @article {gross-zagier, MRKEY = {0833192},
      AUTHOR = {Gross, Benedict H. and Zagier, Don B.},
      TITLE = {Heegner points and derivatives of {$L$}-series},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {84},
      YEAR = {1986},
      NUMBER = {2},
      PAGES = {225--320},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G40 (11F11 11G05 14G10)},
      MRNUMBER = {0833192},
      MRREVIEWER = {Loren D. Olson},
      DOI = {10.1007/BF01388809},
      ZBLNUMBER = {0608.14019},
      }
  • [grothendieck74] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Les Presses de l’Université de Montréal, Montreal, Que., 1974.
    @book {grothendieck74, MRKEY = {0417192},
      AUTHOR = {Grothendieck, Alexandre},
      TITLE = {Groupes de {B}arsotti-{T}ate et cristaux de {D}ieudonné},
      NOTE = {S{é}minaire de Math{é}matiques Sup{é}rieures, No. 45 ({É}t{é},
      1970)},
      PUBLISHER = {Les Presses de l'Université de Montréal, Montreal, Que.},
      YEAR = {1974},
      PAGES = {155},
      MRCLASS = {14L05 (14F30)},
      MRNUMBER = {0417192},
      MRREVIEWER = {Robert A. Morris},
      ZBLNUMBER = {0331.14021},
      }
  • [howard-yangA] B. Howard and T. Yang, "Singular moduli refined," in Arithmetic Geometry and Automorphic Forms, Beijing and International Press, Boston: Higher Education Press, 2011, vol. 19, pp. 367-406.
    @incollection{howard-yangA,
      author={Howard, B. and Yang, T.},
      TITLE={Singular moduli refined},
      BOOKTITLE={Arithmetic Geometry and Automorphic Forms},
      SERIES={Adv. Lectures in Math.},
      VOLUME={19},
      PAGES={367--406},
      YEAR={2011},
      PUBLISHER={Higher Education Press},
      ADDRESS={Beijing and International Press, Boston},
     }
  • [kottwitz92] Go to document R. E. Kottwitz, "Points on some Shimura varieties over finite fields," J. Amer. Math. Soc., vol. 5, iss. 2, pp. 373-444, 1992.
    @article {kottwitz92, MRKEY = {1124982},
      AUTHOR = {Kottwitz, Robert E.},
      TITLE = {Points on some {S}himura varieties over finite fields},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {5},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {373--444},
      ISSN = {0894-0347},
      MRCLASS = {11G18 (11G25 14G10 14G35)},
      MRNUMBER = {1124982},
      MRREVIEWER = {Min Ho Lee},
      DOI = {10.2307/2152772},
      ZBLNUMBER = {0796.14014},
      }
  • [kramer] Go to document N. Krämer, "Local models for ramified unitary groups," Abh. Math. Sem. Univ. Hamburg, vol. 73, pp. 67-80, 2003.
    @article {kramer, MRKEY = {2028507},
      AUTHOR = {Kr{ä}mer, N.},
      TITLE = {Local models for ramified unitary groups},
      JOURNAL = {Abh. Math. Sem. Univ. Hamburg},
      FJOURNAL = {Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg},
      VOLUME = {73},
      YEAR = {2003},
      PAGES = {67--80},
      ISSN = {0025-5858},
      CODEN = {AMHAAJ},
      MRCLASS = {14G35 (11G18)},
      MRNUMBER = {2028507},
      MRREVIEWER = {Chia-Fu Yu},
      DOI = {10.1007/BF02941269},
      ZBLNUMBER = {1053.14024},
     }
  • [kudla96] Go to document S. S. Kudla, Notes on the local theta correspondence, 1996.
    @misc{kudla96,
      author={Kudla, Stephen S.},
      TITLE = {Notes on the local theta correspondence},
      URL = {http://www.math.toronto.edu/~skudla/castle.pdf},
      YEAR={1996},
     }
  • [kudla97] Go to document S. S. Kudla, "Central derivatives of Eisenstein series and height pairings," Ann. of Math., vol. 146, iss. 3, pp. 545-646, 1997.
    @article {kudla97, MRKEY = {1491448},
      AUTHOR = {Kudla, Stephen S.},
      TITLE = {Central derivatives of {E}isenstein series and height pairings},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {146},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {545--646},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11F46 (11F30 11G18 14G40)},
      MRNUMBER = {1491448},
      MRREVIEWER = {Martin L. Karel},
      DOI = {10.2307/2952456},
      ZBLNUMBER = {0990.11032},
      }
  • [kudla-tate] S. S. Kudla, "Tate’s thesis," in An Introduction to the Langlands Program, Boston, MA: Birkhäuser, 2003, pp. 109-131.
    @incollection {kudla-tate, MRKEY = {1990377},
      AUTHOR = {Kudla, Stephen S.},
      TITLE = {Tate's thesis},
      BOOKTITLE = {An Introduction to the {L}anglands Program},
      VENUE={{J}erusalem, 2001},
      PAGES = {109--131},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2003},
      MRCLASS = {11R56 (11F70 11R42)},
      MRNUMBER = {1990377},
      ZBLNUMBER = {1111.11315},
      }
  • [kudla04b] Go to document S. S. Kudla, "Special cycles and derivatives of Eisenstein series," in Heegner Points and Rankin $L$-series, Cambridge: Cambridge Univ. Press, 2004, vol. 49, pp. 243-270.
    @incollection {kudla04b, MRKEY = {2083214},
      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},
      ADDRESS = {Cambridge},
      YEAR = {2004},
      MRCLASS = {11G18 (11F37 11F46 11F67 14G40)},
      MRNUMBER = {2083214},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.1017/CBO9780511756375.009},
      ZBLNUMBER = {1073.11042},
      }
  • [kudla-rallis] Go to document S. S. Kudla and S. Rallis, "On the Weil-Siegel formula," J. Reine Angew. Math., vol. 387, pp. 1-68, 1988.
    @article {kudla-rallis, MRKEY = {0946349},
      AUTHOR = {Kudla, Stephen S. and Rallis, Stephen},
      TITLE = {On the {W}eil-{S}iegel formula},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {387},
      YEAR = {1988},
      PAGES = {1--68},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11F27 (11E45 11F46)},
      MRNUMBER = {0946349},
      MRREVIEWER = {Solomon Friedberg},
      DOI = {10.1515/crll.1988.391.65},
      ZBLNUMBER = {0644.10021},
      }
  • [KRunitaryI] Go to document S. S. Kudla and M. Rapoport, "Special cycles on unitary Shimura varieties I. Unramified local theory," Invent. Math., vol. 184, iss. 3, pp. 629-682, 2011.
    @article {KRunitaryI, MRKEY = {2800697},
      AUTHOR = {Kudla, Stephen S. and Rapoport, Michael},
      TITLE = {Special cycles on unitary {S}himura varieties {I}. {U}nramified local theory},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {184},
      YEAR = {2011},
      NUMBER = {3},
      PAGES = {629--682},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14G35 (11G18)},
      MRNUMBER = {2800697},
      DOI = {10.1007/s00222-010-0298-z},
      ZBLNUMBER={1229.14020},
     }
  • [KRunitaryII] S. S. Kudla and M. Rapoport, Special cycles on unitary Shimura varieties II. Global theory, 2011.
    @misc {KRunitaryII, MRKEY = {2800697},
      AUTHOR = {Kudla, Stephen S. and Rapoport, Michael},
      TITLE = {Special cycles on unitary {S}himura varieties {II}. {G}lobal theory},
      YEAR={2011},
      arXiv = {0912.3758},
     }
  • [kudla99b] Go to document S. S. Kudla, M. Rapoport, and T. Yang, "On the derivative of an Eisenstein series of weight one," Internat. Math. Res. Notices, iss. 7, pp. 347-385, 1999.
    @article {kudla99b, MRKEY = {1683308},
      AUTHOR = {Kudla, Stephen S. and Rapoport, Michael and Yang, Tonghai},
      TITLE = {On the derivative of an {E}isenstein series of weight one},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {1999},
      NUMBER = {7},
      PAGES = {347--385},
      ISSN = {1073-7928},
      MRCLASS = {11F66 (11F30 11F37 11F67 11G40 14G40)},
      MRNUMBER = {1683308},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.1155/S1073792899000185},
      ZBLNUMBER = {1009.11032},
      }
  • [kudla04a] 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 {kudla04a, MRKEY = {2059224},
      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},
      }
  • [KRY] S. S. Kudla, M. Rapoport, and T. Yang, Modular Forms and Special Cycles on Shimura Curves, Princeton, NJ: Princeton Univ. Press, 2006, vol. 161.
    @book {KRY, MRKEY = {2220359},
      AUTHOR = {Kudla, Stephen S. and Rapoport, Michael and Yang, Tonghai},
      TITLE = {Modular Forms and Special Cycles on {S}himura Curves},
      SERIES = {Ann. of Math. Studies},
      VOLUME = {161},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {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},
      ZBLNUMBER = {1157.11027} ,}
  • [kw-lan] Go to document K. Lan, Arithmetic compactifications of PEL-type Shimura varieties, ProQuest LLC, Ann Arbor, MI, 2008.
    @book {kw-lan, MRKEY = {2711676},
      AUTHOR = {Lan, Kai-Wen},
      TITLE = {Arithmetic compactifications of {PEL}-type {S}himura varieties},
      NOTE = {Ph.D. thesis, Harvard University},
      PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
      YEAR = {2008},
      PAGES = {1077},
      ISBN = {978-0549-61380-0},
      MRCLASS = {Thesis},
      MRNUMBER = {2711676},
      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:3312427},
      }
  • [larsen] M. J. Larsen, "Arithmetic compactification of some Shimura surfaces," in The Zeta Functions of Picard Modular Surfaces, Montreal, QC: Univ. Montréal, 1992, pp. 31-45.
    @incollection {larsen, MRKEY = {1155225},
      AUTHOR = {Larsen, Michael J.},
      TITLE = {Arithmetic compactification of some {S}himura surfaces},
      BOOKTITLE = {The Zeta Functions of {P}icard Modular Surfaces},
      PAGES = {31--45},
      PUBLISHER = {Univ. Montréal},
      ADDRESS = {Montreal, QC},
      YEAR = {1992},
      MRCLASS = {14G35 (14J10 14K10)},
      MRNUMBER = {1155225},
      MRREVIEWER = {V. Kumar Murty},
      ZBLNUMBER = {0760.14009},
      }
  • [messing72] Go to document W. Messing, The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Schemes, New York: Springer-Verlag, 1972, vol. 264.
    @book {messing72, MRKEY = {0347836},
      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},
      ADDRESS = {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},
     }
  • [messing07] W. Messing, "Travaux de Zink," in Séminaire Bourbaki. Vol. 2005/2006, , 2007, vol. 311, p. exp. no. 964, ix, 341-364.
    @incollection {messing07, MRKEY = {2359049},
      AUTHOR = {Messing, William},
      TITLE = {Travaux de {Z}ink},
      BOOKTITLE = {S{é}minaire Bourbaki. Vol. 2005/2006},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {311},
      YEAR = {2007},
      PAGES = {Exp. No. 964, ix, 341--364},
      ISSN = {0303-1179},
      ISBN = {978-2-85629-230-3},
      MRCLASS = {14L05 (14B12 14F30)},
      MRNUMBER = {2359049},
      MRREVIEWER = {R. T. Hoobler},
      ZBLNUMBER = {1197.14050},
     }
  • [ARGOS-9] V. Meusers, "Canonical and quasi-canonical liftings in the split case," Astérisque, iss. 312, pp. 87-98, 2007.
    @article {ARGOS-9, MRKEY = {2340373},
      AUTHOR = {Meusers, Volker},
      TITLE = {Canonical and quasi-canonical liftings in the split case},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      NUMBER = {312},
      YEAR = {2007},
      PAGES = {87--98},
      ISSN = {0303-1179},
      ISBN = {978-2-85629-231-0},
      MRCLASS = {11G15 (14K22 14L05)},
      MRNUMBER = {2340373},
      MRREVIEWER = {Reinier Br{ö}ker},
      ZBLNUMBER = {1223.14051},
      }
  • [miller09] Go to document A. Miller, "Compactifications of universal abelian threefolds with CM," Geom. Dedicata, vol. 143, pp. 155-179, 2009.
    @article {miller09, MRKEY = {2576300},
      AUTHOR = {Miller, Andrea},
      TITLE = {Compactifications of universal abelian threefolds with {CM}},
      JOURNAL = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {143},
      YEAR = {2009},
      PAGES = {155--179},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {14G35 (14D06 14K10 14M27)},
      MRNUMBER = {2576300},
      MRREVIEWER = {Jae-Hyun Yang},
      DOI = {10.1007/s10711-009-9380-3},
      ZBLNUMBER = {1198.14032},
      }
  • [oort74] Go to document F. Oort, "Subvarieties of moduli spaces," Invent. Math., vol. 24, pp. 95-119, 1974.
    @article {oort74, MRKEY = {0424813},
      AUTHOR = {Oort, Frans},
      TITLE = {Subvarieties of moduli spaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {24},
      YEAR = {1974},
      PAGES = {95--119},
      ISSN = {0020-9910},
      MRCLASS = {14D20 (14H10 14K10)},
      MRNUMBER = {0424813},
      MRREVIEWER = {H. Popp},
      ZBLNUMBER = {0259.14011},
      DOI = {10.1007/BF01404301},
     }
  • [pappas00] G. Pappas, "On the arithmetic moduli schemes of PEL Shimura varieties," J. Algebraic Geom., vol. 9, iss. 3, pp. 577-605, 2000.
    @article {pappas00, MRKEY = {1752014},
      AUTHOR = {Pappas, Georgios},
      TITLE = {On the arithmetic moduli schemes of {PEL} {S}himura varieties},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {9},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {577--605},
      ISSN = {1056-3911},
      MRCLASS = {14G35 (11G18)},
      MRNUMBER = {1752014},
      MRREVIEWER = {Conjeeveram S. Rajan},
      ZBLNUMBER = {0978.14023},
      }
  • [rapoport78] Go to document M. Rapoport, "Compactifications de l’espace de modules de Hilbert-Blumenthal," Compositio Math., vol. 36, iss. 3, pp. 255-335, 1978.
    @article {rapoport78, MRKEY = {0515050},
      AUTHOR = {Rapoport, M.},
      TITLE = {Compactifications de l'espace de modules de {H}ilbert-{B}lumenthal},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {36},
      YEAR = {1978},
      NUMBER = {3},
      PAGES = {255--335},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14D20 (10D20)},
      MRNUMBER = {0515050},
      MRREVIEWER = {Jean Giraud},
      URL = {http://www.numdam.org/item?id=CM_1978__36_3_255_0},
      ZBLNUMBER = {0386.14006},
      }
  • [serre00] J. Serre, Local Algebra, New York: Springer-Verlag, 2000.
    @book {serre00, MRKEY = {1771925},
      AUTHOR = {Serre, Jean-Pierre},
      TITLE = {Local Algebra},
      SERIES = {Springer Monogr. Math.},
      NOTE = {translated from the French by CheeWhye Chin and revised by the author},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2000},
      PAGES = {xiv+128},
      ISBN = {3-540-66641-9},
      MRCLASS = {13-01 (13D05 13H05 13H15)},
      MRNUMBER = {1771925},
      ZBLNUMBER = {0959.13010},
      }
  • [soule92] Go to document C. Soulé, Lectures on Arakelov Geometry, Cambridge: Cambridge Univ. Press, 1992, vol. 33.
    @book{soule92,
      author={Soul{é},
      C.},
      TITLE={Lectures on {A}rakelov Geometry},
      SERIES={Cambridge Studies in Adv. Math.},
      VOLUME={33},
      YEAR={1992},
      PUBLISHER={Cambridge Univ. Press},
      ADDRESS={Cambridge},
      NOTE={with the collaboration of D.~Abramovich, J.-F.~Burnol, and J.~Kramer},
      MRNUMBER = {1208731},
      ZBLNUMBER = {0812.14015},
      DOI = {10.1017/CBO9780511623950},
     }
  • [tate67] J. T. Tate, "$p$-Divisible Groups," in Proc. Conf. Local Fields, New York: Springer-Verlag, 1967, pp. 158-183.
    @incollection {tate67, MRKEY = {0231827},
      AUTHOR = {Tate, J. T.},
      TITLE = {$p$-Divisible Groups},
      BOOKTITLE = {Proc. {C}onf. {L}ocal {F}ields},
      VENUE={{D}riebergen, 1966},
      PAGES = {158--183},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1967},
      MRCLASS = {14.50},
      MRNUMBER = {0231827},
      MRREVIEWER = {M. Miyanishi},
      ZBLNUMBER = {0157.27601},
      }
  • [terstiege] U. Terstiege, Intersections of special cycles on the Shimura variety for GU$(1,2)$, 2010.
    @misc{terstiege,
      author={Terstiege, U.},
      TITLE={Intersections of special cycles on the {S}himura variety for {GU}$(1,2)$},
      NOTE={preprint},
      YEAR={2010},
      ARXIV = {1006.2106},
     }
  • [vistoli] Go to document A. Vistoli, "Intersection theory on algebraic stacks and on their moduli spaces," Invent. Math., vol. 97, iss. 3, pp. 613-670, 1989.
    @article {vistoli, MRKEY = {1005008},
      AUTHOR = {Vistoli, Angelo},
      TITLE = {Intersection theory on algebraic stacks and on their moduli spaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {97},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {613--670},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14C17 (14A20 14D20)},
      MRNUMBER = {1005008},
      MRREVIEWER = {Steven E. Landsburg},
      DOI = {10.1007/BF01388892},
      ZBLNUMBER = {0694.14001},
      }
  • [ARGOS-8] S. Wewers, Canonical and Quasi-Canonical Liftings, , 2007, vol. 312.
    @book {ARGOS-8, MRKEY = {2340372},
      AUTHOR = {Wewers, Stefan},
      TITLE = {Canonical and Quasi-Canonical Liftings},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {312},
      YEAR = {2007},
      PAGES = {67--86},
      ISSN = {0303-1179},
      ISBN = {978-2-85629-231-0},
      MRCLASS = {14L05 (14K22)},
      MRNUMBER = {2340372},
      MRREVIEWER = {Alan Koch},
      ZBLNUMBER = {1156.14323},
      }
  • [yang05] T. Yang, "CM number fields and modular forms," Pure Appl. Math. Q., vol. 1, iss. 2, part 1, pp. 305-340, 2005.
    @article {yang05, MRKEY = {2194727},
      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, part 1},
      PAGES = {305--340},
      ISSN = {1558-8599},
      MRCLASS = {11F41 (11R80)},
      MRNUMBER = {2194727},
      MRREVIEWER = {Jens Funke},
      ZBLNUMBER = {1146.11028},
      }
  • [zink02] T. Zink, "The display of a formal $p$-divisible group," in Cohomologies $p$-Adiques et Applications Arithmétiques, I, , 2002, vol. 278, pp. 127-248.
    @incollection {zink02, MRKEY = {1922825},
      AUTHOR = {Zink, Thomas},
      TITLE = {The display of a formal {$p$}-divisible group},
      BOOKTITLE = {Cohomologies $p$-Adiques et Applications Arithm{é}tiques, I},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {278},
      YEAR = {2002},
      PAGES = {127--248},
      ISSN = {0303-1179},
      MRCLASS = {14L05 (14F30)},
      MRNUMBER = {1922825},
      MRREVIEWER = {Oliver B{ü}ltel},
      ZBLNUMBER = {1008.14008},
      }

Authors

Benjamin Howard

Department of Mathematics, Boston College, Chestnut Hill, MA 92467