Annals of Global Analysis and Geometry

, Volume 55, Issue 3, pp 417–441 | Cite as

On vector-valued automorphic forms on bounded symmetric domains

  • Nadia Alluhaibi
  • Tatyana BarronEmail author


We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in \({\mathbb {C}}^n\), we provide estimates for the norms of these automorphic forms and we find asymptotics of the norms (as the weight goes to infinity) for a class of totally real submanifolds. We give an example of a CR submanifold of the ball, for which the norms of the associated automorphic forms have a different asymptotic behaviour.


Holomorphic automorphic forms Poincaré series Spanning set Domain Canonical bundle Bergman kernel Complex hyperbolic space Submanifold Asymptotics 

Mathematics Subject Classification

32N15 53C99 



We are thankful to A. Dhillon, Y. Karshon, M. Pinsonnault, E. Schippers, A. Uribe, and N. Yui for related discussions. We acknowledge the referee’s efforts.


  1. 1.
    Ali, S., Englis, M.: Matrix-valued Berezin–Toeplitz quantization. J. Math. Phys. 48(5), 053504 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alluhaibi, N.: On vector-valued automorphic forms on bounded symmetric domains. Ph.D. Thesis, University of Western Ontario (2017)Google Scholar
  3. 3.
    Baily, W.: Introductory Lectures on Automorphic Forms. Princeton University Press, Princeton (1973)zbMATHGoogle Scholar
  4. 4.
    Barron, T.: Quantization and automorphic forms. Contemp. Math. 583, 211–219 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barron, T.: Closed geodesics and pluricanonical sections on ball quotients. arxiv:1808.01245
  6. 6.
    Bell, D.: Poincaré series representations of automorphic forms. Ph.D. Thesis, Brown University (1967)Google Scholar
  7. 7.
    Bleistein, N., Handelsman, R.: Asymptotic Expansions of Integrals. Holt, Rinehart, Winston (1975)zbMATHGoogle Scholar
  8. 8.
    Borel, A.: Introduction to automorphic forms. In: Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposium in Pure Mathematics, Boulder, CO, 1965). AMS, Providence, pp. 199–210 (1966)Google Scholar
  9. 9.
    Borthwick, D., Paul, T., Uribe, A.: Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122(2), 359–402 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Burns, D., Guillemin, V., Wang, Z.: Stability functions. Geom. Funct. Anal. 19(5), 1258–1295 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cléry, F., van der Geer, G.: Generators for modules of vector-valued Picard modular forms. Nagoya Math. J. 212, 19–57 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    de Bruijn, N.: Asymptotic methods in analysis. Second edition. Bibliotheca Mathematica, Vol. IV. North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen (1961)Google Scholar
  13. 13.
    Debernardi, M., Paoletti, R.: Equivariant asymptotics for Bohr–Sommerfeld Lagrangian submanifolds. Commun. Math. Phys. 267(1), 227–263 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Foth, T.: Poincaré series on bounded symmetric domains. Proc. AMS 135(10), 3301–3308 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Foth, T.: Legendrian tori and the semi-classical limit. Diff. Geom. Appl. 26(1), 63–74 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Foth, T., Katok, S.: Spanning sets for automorphic forms and dynamics of the frame flow on complex hyperbolic spaces. Ergod. Theory Dyn. Syst. 21(4), 1071–1099 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Foth, T., Katok, S.: Appendix to S. Katok. Livshitz theorem for the unitary frame flow. Ergodic Theory Dyn. Syst. 24(1), 127–140; pp. 137–140 (2004)Google Scholar
  18. 18.
    Freitag, E., Manni, R.: Vector valued modular forms on three dimensional ball. arxiv:1404.3057
  19. 19.
    Goldman, W.: Complex Hyperbolic Geometry. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1999)Google Scholar
  20. 20.
    Gorodentsev, A., Tyurin, A.: Abelian Lagrangian algebraic geometry. Izv. Math. 65(3), 437–467 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guillemin, V., Uribe, A., Wang, Z.: Semiclassical states associated with isotropic submanifolds of phase space. Lett. Math. Phys. 106(12), 1695–1728 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hsu, L.: On the asymptotic evaluation of a class of multiple integrals involving a parameter. Am. J. Math. 73, 625–634 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ioos, L.: Quantization and isotropic submanifolds. arxiv:1802.09930
  24. 24.
    Jeffrey, L., Weitsman, J.: Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Commun. Math. Phys. 150(3), 593–630 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kato, S.: A dimension formula for a certain space of automorphic forms of SU(p,1). Math. Ann. 266(4), 457–477 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Katok, S.: Closed geodesics, periods and arithmetic of modular forms. Invent. Math. 80(3), 469–480 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Katok, S., Millson, J.: Eichler–Shimura homology, intersection numbers and rational structures on spaces of modular forms. Trans. Am. Math. Soc. 300(2), 737–757 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Princeton University Press, Princeton (1987)CrossRefzbMATHGoogle Scholar
  29. 29.
    Kojima, H.: The formula for the dimension of the spaces of vector-valued holomorphic automorphic forms on the unitary group SU(1, p). Kyushu J. Math. 51(1), 57–76 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kollár, J.: Shafarevich Maps and Automorphic Forms. Princeton University Press, Princeton (1995)CrossRefzbMATHGoogle Scholar
  31. 31.
    Kudla, S., Millson, J.: Harmonic differentials and closed geodesics on a Riemann surface. Invent. Math. 54(3), 193–211 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lu, Z., Zelditch, S.: Szegö kernels and Poincaré series. J. Anal. Math. 130, 167–184 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ma, X., Marinescu, G.: Exponential estimate for the asymptotics of Bergman kernels. Math. Ann. 362(3–4), 1327–1347 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Narasimhan, M., Seshadri, C.: Holomorphic vector bundles on a compact Riemann surface. Math. Ann. 155, 69–80 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Paoletti, R.: A note on scaling asymptotics for Bohr–Sommerfeld Lagrangian submanifolds. Proc. Am. Math. Soc. 136(11), 4011–4017 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Parker, J.: Complex hyperbolic lattices. In: Discrete Groups and Geometric Structures, 1–42, Contemporary Mathematics, 501. American Mathematical Society, Providence, RI (2009)Google Scholar
  37. 37.
    Range, R.M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  38. 38.
    Rudin, W.: Function Theory in the Unit Ball of \(\mathbb{C}^n\). Springer, New York, Berlin (1980)CrossRefzbMATHGoogle Scholar
  39. 39.
    Selberg, A.: Automorphic functions and integral operators. In: Collected Papers, vol. I. Springer, Berlin, pp. 464–468 (1989)Google Scholar
  40. 40.
    Selberg, A.: Recent developments in the theory of discontinuous groups of motions of symmetric spaces. In: Collected Papers, vol. I. Springer, Berlin, pp. 546–567 (1989)Google Scholar
  41. 41.
    Tong, Y., Wang, S.: Theta functions defined by geodesic cycles in quotients of SU(p,1). Invent. Math. 71(3), 467–499 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wu, J., Wang, X.: Poincaré series and very ampleness criterion for pluri-canonical bundles. arxiv:1504.00081
  43. 43.
    Wong, R.: Asymptotic Approximations of Integrals. Academic Press, Boston (1989)zbMATHGoogle Scholar
  44. 44.
    Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005)zbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Science and Arts College, Rabigh CampusKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of MathematicsUniversity of Western OntarioLondonCanada

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