Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Poincaré series on bounded symmetric domains


Author: Tatyana Foth
Journal: Proc. Amer. Math. Soc. 135 (2007), 3301-3308
MSC (2000): Primary 32N10; Secondary 32N05, 32N15
Published electronically: June 21, 2007
MathSciNet review: 2322762
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that any holomorphic automorphic form of sufficiently large weight on an irreducible bounded symmetric domain in $ {\mathbb{C}}^n$, $ n>1$, is the Poincaré series of a polynomial in $ z_1$,...,$ z_n$ and give an upper bound for the degree of this polynomial. We also give an explicit construction of a basis in the space of holomorphic automorphic forms.


References [Enhancements On Off] (What's this?)

  • 1. W. Barth and R. Narasimhan (eds.), Several complex variables. VI, Encyclopaedia of Mathematical Sciences, vol. 69, Springer-Verlag, Berlin, 1990. Complex manifolds; A translation of Sovremennye problemy matematiki. Fundamentalnye napravleniya, Tom 69, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. MR 1095088
  • 2. Walter L. Baily Jr., Introductory lectures on automorphic forms, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1973. Kan\cflex o Memorial Lectures, No. 2; Publications of the Mathematical Society of Japan, No. 12. MR 0369750
  • 3. Lipman Bers, Completeness theorems for Poincaré series in one variable, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 88–100. MR 0132829
  • 4. Armand Borel, Introduction to automorphic forms, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 199–210. MR 0207650
  • 5. Henri Cartan, Fonctions automorphes et séries de Poincaré, J. Analyse Math. 6 (1958), 169–175. MR 0099450
  • 6. I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Translated from the Russian by K. A. Hirsch, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR 0233772
  • 7. Dennis A. Hejhal, Monodromy groups and Poincaré series, Bull. Amer. Math. Soc. 84 (1978), no. 3, 339–376. MR 0492237, 10.1090/S0002-9904-1978-14469-3
  • 8. Dennis A. Hejhal, Kernel functions, Poincaré series, and LVA, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 173–201. MR 1759678, 10.1090/conm/256/04005
  • 9. L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translated from the Russian by Leo Ebner and Adam Korányi, American Mathematical Society, Providence, R.I., 1963. MR 0171936
  • 10. Masa-Nori Ishida, The irregularities of Hirzebruch’s examples of surfaces of general type with 𝑐²₁=3𝑐₂, Math. Ann. 262 (1983), no. 3, 407–420. MR 692865, 10.1007/BF01456018
  • 11. G. Khenkin, The method of integral representations in complex analysis, in Several complex variables I, Encycl. Math. Sci., Vol. 7, Springer-Verlag, 1990; pp. 19-116.
  • 12. János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR 1341589
  • 13. Irwin Kra, Automorphic forms and Kleinian groups, W. A. Benjamin, Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. MR 0357775
  • 14. Irwin Kra, On the vanishing of and spanning sets for Poincaré series for cusp forms, Acta Math. 153 (1984), no. 1-2, 47–116. MR 744999, 10.1007/BF02392375
  • 15. Irwin Kra, Cusp forms associated to loxodromic elements of Kleinian groups, Duke Math. J. 52 (1985), no. 3, 587–625. MR 808095, 10.1215/S0012-7094-85-05232-9
  • 16. Irwin Kra, Bases for cusp forms for quasi-Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 289–298. MR 802490, 10.5186/aasfm.1985.1032
  • 17. Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625
  • 18. Yozô Matsushima, On the first Betti number of compact quotient spaces of higher-dimensional symmetric spaces, Ann. of Math. (2) 75 (1962), 312–330. MR 0158406
  • 19. Yozô Matsushima, On Betti numbers of compact, locally sysmmetric Riemannian manifolds, Osaka Math. J. 14 (1962), 1–20. MR 0141138
  • 20. H. Petersson, Über Weierstrasspunkte und die expliziten Darstellungen der automorphen Formen von reeller Dimension, Math. Z. 52 (1949), 32–59 (German). MR 0033356
  • 21. I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Translated from the Russian. Mathematics and Its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0252690
  • 22. A. Selberg, Automorphic functions and integral operators, in Collected papers, vol. I, 464-468.
  • 23. Harald Upmeier, Toeplitz operators and index theory in several complex variables, Operator Theory: Advances and Applications, vol. 81, Birkhäuser Verlag, Basel, 1996. MR 1384981
  • 24. Joseph A. Wolf, Completeness of Poincaré series for automorphic cohomology, Ann. of Math. (2) 109 (1979), no. 3, 545–567. MR 534762, 10.2307/1971225

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32N10, 32N05, 32N15

Retrieve articles in all journals with MSC (2000): 32N10, 32N05, 32N15


Additional Information

Tatyana Foth
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
Email: tfoth@uwo.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08862-4
Received by editor(s): October 6, 2005
Received by editor(s) in revised form: July 25, 2006
Published electronically: June 21, 2007
Additional Notes: The research of this author was supported in part by NSF grant DMS-0204154
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2007 American Mathematical Society