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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The kernel of the Poincaré series operator
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by Thomas A. Metzger PDF
Proc. Amer. Math. Soc. 76 (1979), 289-292 Request permission

Abstract:

By modifying a proof of Ljan, a natural basis for the kernel of the Poincaré series operator in the Bers space can be given. The basic idea behind the proof also extends to give such a basis in the case of a general Kleinian group and a discontinuous group acting on certain domains in ${{\mathbf {C}}^n}$.
References
  • 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
  • Lipman Bers, Automorphic forms and Poincaré series for infinitely generated Fuchsian groups, Amer. J. Math. 87 (1965), 196–214. MR 174737, DOI 10.2307/2373231
  • David Drasin, Cusp forms and Poincaré series, Amer. J. Math. 90 (1968), 356–365. MR 229818, DOI 10.2307/2373532
  • Clifford J. Earle, Some remarks on Poincaré series, Compositio Math. 21 (1969), 167–176. MR 249664
  • Irwin Kra, Automorphic forms and Kleinian groups, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR 0357775
  • G. M. Ljan, On the kernel of Poincaré’s $\theta$ operator, Soviet Math. Dokl. 17 (1976), 1283-1285.
  • 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
  • H. Poincare, Mémoire sur les fonctions fuchsiennes, Acta Math. 1 (1882), no. 1, 193–294 (French). MR 1554584, DOI 10.1007/BF02391845
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 76 (1979), 289-292
  • MSC: Primary 30F35
  • DOI: https://doi.org/10.1090/S0002-9939-1979-0537090-0
  • MathSciNet review: 537090