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Intersection theory on Shimura surfaces

Part of: Discontinuous groups and automorphic forms Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  01 March 2009

Benjamin Howard
Benjamin Howard*
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA, 02467, USA (email: howardbe@bc.edu)
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Abstract

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Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycle classes constructed by Kudla, Rapoport and Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

Keywords

Shimura varietiesArakelov theoryKudla’s program

MSC classification

Secondary: 14G35: Modular and Shimura varieties 14G40: Arithmetic varieties and schemes; Arakelov theory; heights 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 423 - 475
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Andreatta, F. and Goren, E. Z., Geometry of Hilbert modular varieties over totally ramified primes, Int. Math. Res. Not. 33 (2003), 1786–1835.Google Scholar
[2]Andreatta, F. and Goren, E. Z., Hilbert modular varieties of low dimension, Geometric Aspects of Dwork Theory. vol. I, II (Walter de Gruyter GmbH & Co. KG, Berlin, 2004), 113–175.Google Scholar
[3]Andreatta, F. and Goren, E. Z., Hilbert modular forms: mod p and p-adic aspects, Mem. Amer. Math. Soc. 173 (2005), vi+100.Google Scholar
[4]Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21 (Springer-Verlag, Berlin, 1990).Google Scholar
[5]Boutot, J.-F., Uniformisation p-adique des variétés de Shimura, Astérisque 245 (1997), 307–322.Google Scholar
[6]Boutot, J.-F., Breen, L., Gérardin, P., Giraud, J., Labesse, J.-P., Milne, J. and Soulé, C., Variétés de Shimura et fonctions L, Publications Mathématiques de l’Université Paris VII [Mathematical Publications of the University of Paris VII], vol. 6 (Université de Paris VII U.E.R. de Mathématiques, Paris, 1979).Google Scholar
[7]Boutot, J.-F. and Carayol, H., Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel’d, Astérisque 196–197 (1991), 45–158. (Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).)Google Scholar
[8]Boutot, J.-F. and Zink, T., The p-adic uniformization of Shimura curves. Preprint, http://www.mathematik.uni-bielefeld.de/∼zink.Google Scholar
[9]Bruinier, Jan H., Burgos Gil, J. I. and Kühn, U., Borcherds products and arithmetic intersection theory on Hilbert modular surfaces, Duke Math. J. 139 (2007), 1–88.Google Scholar
[10]Burgos Gil, J. I., Kramer, J. and Kühn, U., Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), 1–172.Google Scholar
[11]Buzzard, K., Integral models of certain Shimura curves, Duke Math. J. 87 (1997), 591–612.Google Scholar
[12]Conrad, B., Gross-Zagier revisited, in Heegner Points and Rankin L-series, Mathematical Science Research Institute Publications, vol. 49 (Cambridge University Press, Cambridge, 2004), 67–163. (With an appendix by W. R. Mann.)Google Scholar
[13]Deligne, P. and Pappas, G., Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math. 90 (1994), 59–79.Google Scholar
[14]Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S. L., Nitsure, N. and Vistoli, A., Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs, vol. 123 (American Mathematical Society, Providence, RI, 2005).Google Scholar
[15]Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, second edition (Springer, Berlin, 1998).Google Scholar
[16]Gillet, H., Intersection theory on algebraic stacks and Q-varieties, J. Pure Appl. Algebra 34 (1984), 193–240.Google Scholar
[17]Gillet, H. and Soulé, C., Intersection theory using Adams operations, Invent. Math. 90 (1987), 243–277.Google Scholar
[18]Gillet, H. and Soulé, C., Arithmetic intersection theory, Publ. Math. Inst. Hautes Études Sci. 72 (1990), 93–174.Google Scholar
[19]Goren, E. Z., Lectures on Hilbert modular varieties and modular forms, CRM Monograph Series, vol. 14 (American Mathematical Society, Providence, RI, 2002). (With the assistance of Marc-Hubert Nicole.)Google Scholar
[20]Goren, E. Z. and Oort, F., Stratifications of Hilbert modular varieties, J. Algebraic Geom. 9 (2000), 111–154.Google Scholar
[21]Gubler, W., Moving lemma for K 1-chains, J. Reine Angew. Math. 548 (2002), 1–19.Google Scholar
[22]Hida, H., p-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics (Springer-Verlag, New York, 2004).Google Scholar
[23]Hida, H., Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs (The Clarendon Press; Oxford University Press, Oxford, 2006).Google Scholar
[24]Howard, B., Intersection theory on Shimura surfaces II. Preprint.Google Scholar
[25]Kottwitz, R., Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.Google Scholar
[26]Kudla, S., Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997), 545–646.Google Scholar
[27]Kudla, S., Special cycles and derivatives of Eisenstein series, in Heegner points and Rankin L-series, Mathematical Science Research Institute Publications, vol. 49 (Cambridge University Press, Cambridge, 2004), 243–270.Google Scholar
[28]Kudla, S. and Rapoport, M., Arithmetic Hirzebruch-Zagier cycles, J. Reine Angew. Math. 515 (1999), 155–244.Google Scholar
[29]Kudla, S. and Rapoport, M., Height pairings on Shimura curves and p-adic uniformization, Invent. Math. 142 (2000), 153–223.Google Scholar
[30]Kudla, S., Rapoport, M. and Yang, T., Derivatives of Eisenstein series and Faltings heights, Composito Math. 140 (2004), 887–951.Google Scholar
[31]Kudla, S., Rapoport, M. and Yang, T., Modular forms and special cycles on Shimura curves, Annals of Mathematics Studies, vol. 161 (Princeton University Press, Princeton, NJ, 2006).Google Scholar
[32]Lam, T. Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 2005).Google Scholar
[33]Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39 (Springer, Berlin, 2000).Google Scholar
[34]Liu, Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford Science Publications, Oxford University Press, Oxford, 2002). (Translated from the French by Reinie Erné.)Google Scholar
[35]Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 second edition (Cambridge University Press, Cambridge, 1989). (Translated from the Japanese by M. Reid.)Google Scholar
[36]Milne, J., Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in Automorphic forms, Shimura varieties, and L-functions, vol. I, (Ann Arbor, MI, 1988, Perspectives in Mathematics, vol. 10 (Academic Press, Boston, MA, 1990), 283–414.Google Scholar
[37]Milne, J., Introduction to Shimura varieties, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematical Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 265–378.Google Scholar
[38]Milne, J. S., Points on Shimura varieties mod p, in Automorphic forms, representations and L-functions, Oregon State University, Corvallis, OR, 1977, Part 2, Proceedings of Symposia in Pure Mathematics, XXXIII (American Mathematical Society, Providence, RI, 1979), 165–184.Google Scholar
[39]Mumford, D., Geometric invariant theory, in Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34 (Springer, Berlin, 1965).Google Scholar
[40]Pappas, G., Arithmetic models for Hilbert modular varieties, Compositio Math. 98 (1995), 43–76.Google Scholar
[41]Rapoport, M., Compactifications de l’espace de modules de Hilbert–Blumenthal, Compositio Math. 36 (1978), 255–335.Google Scholar
[42]Rapoport, M. and Zink, T., Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), 21–101.Google Scholar
[43]Rapoport, M. and Zink, Th., Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141 (Princeton University Press, Princeton, NJ, 1996).Google Scholar
[44]Serre, J.-P., Local algebra, Springer Monographs in Mathematics (Springer, Berlin, 2000). (Translated from the French by CheeWhye Chin and revised by the author.)Google Scholar
[45]Soulé, C., Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33 (Cambridge University Press, Cambridge, 1992). (With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer.)Google Scholar
[46]Stamm, H., On the reduction of the Hilbert–Blumenthal-moduli scheme with Γ0(p)-level structure, Forum Math. 9 (1997), 405–455.Google Scholar
[47]van der Geer, G., Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16 (Springer, Berlin, 1988).Google Scholar
[48]Vistoli, A., Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.Google Scholar
[49]Vollaard, I., On the Hilbert–Blumenthal moduli problem, J. Inst. Math. Jussieu 4 (2005), 653–683.Google Scholar
[50]Wewers, S., Canonical and quasi-canonical liftings, Astérisque 312 (2007), 67–86.Google Scholar
[51]Zink, T., The display of a formal p-divisible group, Astérisque 278 (2002), 127–248.Google Scholar