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Poincaré series forSO(n, 1)

  • Jian-Shu Li
  • I. Piatetski-Shapiro
  • P. Sarnak
Article

Abstract

A theory of Poincaré series is developed for Lobachevsky space of arbitrary dimension. For a general non-uniform lattice a Selberg-Kloosterman zeta function is introduced. It has meromorphic continuation to the plane with poles at the corresponding automorphic spectrum. When the lattice is a unit group of a rational quadratic form, the Selberg-Kloosterman zeta function is computed explicitly in terms of exponential sums. In this way a non-trivial Ramanujan-like bound analogous to “Selberg’s 3/16 bound” is proved in general.

Keywords

Poincaré series Lobachevsky space Selberg-Kloosterman zeta function non-uniform lattices 

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Copyright information

© Indian Academy of Sciences 1987

Authors and Affiliations

  • Jian-Shu Li
    • 1
  • I. Piatetski-Shapiro
  • P. Sarnak
    • 2
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.The Institute for Advanced StudiesThe Hebrew University of JerusalemJerusalemIsrael

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