Green forms and the arithmetic Siegel–Weil formula

  • Luis E. GarciaEmail author
  • Siddarth Sankaran


We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type \(\mathrm {O}(p,2)\) and \(\mathrm {U}(p,1)\), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.



This work was done while L.G. was at the University of Toronto and IHES and S.S. was at the University of Manitoba; the authors thank these institutions for providing excellent working conditions. An early draft, with a full proof of the main identity for non-degenerate Fourier coefficients, was circulated and posted online in January 2018; we are grateful to Daniel Disegni, Gerard Freixas i Montplet, Stephen Kudla and Shouwu Zhang for comments on it, and other helpful conversations. We also thank the referee for their thorough reading and insightful comments and suggestions. L.G. acknowledges financial support from the ERC AAMOT Advanced Grant. S.S. acknowledges financial support from NSERC.


  1. 1.
    Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators, volume 298 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1992). CrossRefGoogle Scholar
  2. 2.
    Bismut, J.-M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. I. Bott–Chern forms and analytic torsion. Comm. Math. Phys. 115(1), 49–78 (1988)
  3. 3.
    Bismut, J.-M., Gillet, H., Soulé, C.: Bott–Chern currents and complex immersions. Duke Math. J. 60(1), 255–284 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bismut, J.-M.: Superconnection currents and complex immersions. Invent. Math. 99(1), 59–113 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bismut, J.-M., Gillet, H., Soulé, C.: Complex immersions and Arakelov geometry. In: The Grothendieck Festschrift, Vol. I, volume 86 of Progr. Math., pp. 249–331. Birkhäuser Boston, Boston (1990)CrossRefGoogle Scholar
  6. 6.
    Bost, J.-B., Gillet, H., Soulé, C.: Heights of projective varieties and positive Green forms. J. Am. Math. Soc. 7(4), 903–1027 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bruinier, J.H., Burgos Gil, J.I., Kühn, U.: Borcherds products and arithmetic intersection theory on Hilbert modular surfaces. Duke Math. J. 139(1), 1–88 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bruinier, J.H., Yang, T.: CM values of automorphic Green functions on orthogonal groups over totally real fields. In: Arithmetic Geometry and Automorphic Forms, volume 19 of Adv. Lect. Math. (ALM), pp. 1–54. Int. Press, Somerville (2011)Google Scholar
  9. 9.
    Ehlen, S., Sankaran, S.: On two arithmetic theta lifts. Compos. Math. 154(10), 2090–2149 (2018). MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gan, W.T., Qiu, Y., Takeda, S.: The regularized Siegel–Weil formula (the second term identity) and the Rallis inner product formula. Invent. Math. 198(3), 739–831 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Garcia, L.E.: Superconnections, theta series, and period domains. Adv. Math. 329, 555–589 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Harris, M., Kudla, S.S.: Arithmetic automorphic forms for the nonholomorphic discrete series of \({\rm GSp}(2)\). Duke Math. J. 66(1), 59–121 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Harris, M., Kudla, S.S., Sweet, W.J.: Theta dichotomy for unitary groups. J. Am. Math. Soc. 9(4), 941–1004 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bruinier, J.H., Yang, T.: Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series. ArXiv e-prints (2018). arXiv:1802.09489
  15. 15.
    Hörmander, L.: The analysis of linear partial differential operators. I. In: Classics in Mathematics. Springer, Berlin (2003). Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. CrossRefGoogle Scholar
  16. 16.
    Hörmann, F.: The geometric and arithmetic volume of Shimura varieties of orthogonal type, volume 35 of CRM Monograph Series. American Mathematical Society, Providence (2014)Google Scholar
  17. 17.
    Ichino, A.: A regularized Siegel–Weil formula for unitary groups. Math. Z. 247(2), 241–277 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ichino, A.: Pullbacks of Saito–Kurokawa lifts. Invent. Math. 162(3), 551–647 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ichino, A.: On the Siegel–Weil formula for unitary groups. Math. Z. 255(4), 721–729 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kashiwara, M., Vergne, M.: On the Segal–Shale–Weil representations and harmonic polynomials. Invent. Math. 44(1), 1–47 (1978). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kudla, S., Rapoport, M.: Special cycles on unitary Shimura varieties II: Global theory. J. Reine Angew. Math. 697, 91–157 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kudla, S.S.: Splitting metaplectic covers of dual reductive pairs. Isr. J. Math. 87(1–3), 361–401 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kudla, S.S.: Algebraic cycles on Shimura varieties of orthogonal type. Duke Math. J. 86(1), 39–78 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kudla, S.S.: Central derivatives of Eisenstein series and height pairings. Ann. Math. (2) 146(3), 545–646 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kudla, S.S.: Integrals of Borcherds forms. Compos. Math. 137(3), 293–349 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kudla, S.S.: Special cycles and derivatives of Eisenstein series. In: Heegner points and Rankin \(L\)-series, volume 49 of Math. Sci. Res. Inst. Publ., pp. 243–270. Cambridge Univ. Press, Cambridge (2004).
  27. 27.
    Kudla, S.S., Millson, J.J.: The theta correspondence and harmonic forms. I. Math. Ann. 274(3), 353–378 (1986). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kudla, S.S., Millson, J.J.: Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables. Inst. Hautes Études Sci. Publ. Math. 71, 121–172 (1990) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kudla, S.S., Rallis, S.: On the Weil–Siegel formula. J. Reine Angew. Math. 387, 1–68 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kudla, S.S., Rallis, S.: Degenerate principal series and invariant distributions. Isr. J. Math. 69(1), 25–45 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kudla, S.S., Rallis, S.: A regularized Siegel–Weil formula: the first term identity. Ann. Math. (2) 140(1), 1–80 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kudla, S.S., Rapoport, M., Yang, T.: Derivatives of Eisenstein series and Faltings heights. Compos. Math. 140(4), 887–951 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kudla, S.S., Rapoport, M., Yang, T.: Modular forms and special cycles on Shimura curves, volume 161 of Annals of Mathematics Studies. Princeton University Press, Princeton (2006)Google Scholar
  34. 34.
    Soo Teck Lee: On some degenerate principal series representations of \({\rm U}(n, n)\). J. Funct. Anal. 126(2), 305–366 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lee, S.T., Zhu, C.-B.: Degenerate principal series and local theta correspondence. Trans. Am. Math. Soc. 350(12), 5017–5046 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liu, Y.: Arithmetic theta lifting and \(L\)-derivatives for unitary groups. I. Algebra Number Theory 5(7), 849–921 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Mathai, V., Quillen, D.: Superconnections, Thom classes, and equivariant differential forms. Topology 25(1), 85–110 (1986). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Milne, J.S.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Automorphic forms, Shimura varieties, and \(L\)-functions, Vol. I (Ann Arbor, MI, 1988), volume 10 of Perspect. Math., pp. 283–414. Academic Press, Boston (1990)Google Scholar
  39. 39.
    Milne, J.S., Suh, J.: Nonhomeomorphic conjugates of connected Shimura varieties. Am. J. Math. 132(3), 731–750 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Quillen, D.: Superconnections and the Chern character. Topology 24(1), 89–95 (1985). MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rallis, S.: On the Howe duality conjecture. Compos. Math. 51(3), 333–399 (1984)
  42. 42.
    Ranga Rao, R.: On some explicit formulas in the theory of Weil representation. Pac. J. Math. 157(2), 335–371 (1993) MathSciNetCrossRefGoogle Scholar
  43. 43.
    Shih, K.: Existence of certain canonical models. Duke Math. J. 45(1), 63–66 (1978) MathSciNetCrossRefGoogle Scholar
  44. 44.
    Shimura, G.: Confluent hypergeometric functions on tube domains. Math. Ann. 260(3), 269–302 (1982). MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Soulé, C.: Lectures on Arakelov geometry, volume 33 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer.
  46. 46.
    Sweet Jr., W. J.: The metaplectic case of the Weil–Siegel formula. ProQuest LLC, Ann Arbor, MI, (1990). Thesis (Ph.D.)–University of Maryland, College Park. URL:
  47. 47.
    Tan, V.: Poles of Siegel Eisenstein series on \({\rm U}(n, n)\). Canad. J. Math. 51(1), 164–175 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Wallach, N.R.: Holomorphic continuation of generalized Jacquet integrals for degenerate principal series. Represent. Theory 10, 380–398 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Wells Jr., R.O.: Differential analysis on complex manifolds, volume 65 of Graduate Texts in Mathematics. Springer, New York, 3rd edn., 2008. With a new appendix by Oscar Garcia-Prada. CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsUniversity of ManitobaWinnipegCanada

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