Poincaré series on bounded symmetric domains

Foth, Tatyana

doi:10.1090/S0002-9939-07-08862-4

Poincaré series on bounded symmetric domains
HTML articles powered by AMS MathViewer

by Tatyana Foth PDF

Proc. Amer. Math. Soc. 135 (2007), 3301-3308 Request permission

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

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. Fundamental′nye napravleniya, Tom 69, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. MR 1095088
Walter L. Baily Jr., Introductory lectures on automorphic forms, Kanô Memorial Lectures, No. 2, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1973. MR 0369750
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
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
Henri Cartan, Fonctions automorphes et séries de Poincaré, J. Analyse Math. 6 (1958), 169–175. MR 99450, DOI 10.1007/BF02790234
I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR 0233772
Dennis A. Hejhal, Monodromy groups and Poincaré series, Bull. Amer. Math. Soc. 84 (1978), no. 3, 339–376. MR 492237, DOI 10.1090/S0002-9904-1978-14469-3
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, DOI 10.1090/conm/256/04005
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR 0171936
Masa-Nori Ishida, The irregularities of Hirzebruch’s examples of surfaces of general type with $c^{2}_{1}=3c_{2}$, Math. Ann. 262 (1983), no. 3, 407–420. MR 692865, DOI 10.1007/BF01456018
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.
János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR 1341589, DOI 10.1515/9781400864195
Irwin Kra, Automorphic forms and Kleinian groups, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR 0357775
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, DOI 10.1007/BF02392375
Irwin Kra, Cusp forms associated to loxodromic elements of Kleinian groups, Duke Math. J. 52 (1985), no. 3, 587–625. MR 808095, DOI 10.1215/S0012-7094-85-05232-9
Irwin Kra, Bases for cusp forms for quasi-Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 289–298. MR 802490, DOI 10.5186/aasfm.1985.1032
Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
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 158406, DOI 10.2307/1970176
Yozô Matsushima, On Betti numbers of compact, locally sysmmetric Riemannian manifolds, Osaka Math. J. 14 (1962), 1–20. MR 141138
H. Petersson, Über Weierstrasspunkte und die expliziten Darstellungen der automorphen Formen von reeller Dimension, Math. Z. 52 (1949), 32–59 (German). MR 33356, DOI 10.1007/BF02230683
I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Mathematics and its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Translated from the Russian. MR 0252690
A. Selberg, Automorphic functions and integral operators, in Collected papers, vol. I, 464-468.
Harald Upmeier, Toeplitz operators and index theory in several complex variables, Operator Theory: Advances and Applications, vol. 81, Birkhäuser Verlag, Basel, 1996. MR 1384981, DOI 10.1016/s0168-9274(96)90019-7
Joseph A. Wolf, Completeness of Poincaré series for automorphic cohomology, Ann. of Math. (2) 109 (1979), no. 3, 545–567. MR 534762, DOI 10.2307/1971225