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Poincaré series forSO(n, 1)

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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.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, 02139, Cambridge, Massachusetts, USA

    Jian-Shu Li

  2. The Institute for Advanced Studies, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    P. Sarnak

Authors
  1. Jian-Shu Li
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  2. I. Piatetski-Shapiro
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  3. P. Sarnak
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Li, JS., Piatetski-Shapiro, I. & Sarnak, P. Poincaré series forSO(n, 1). Proc. Indian Acad. Sci. (Math. Sci.) 97, 231–237 (1987). https://doi.org/10.1007/BF02837825

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  • DOI: https://doi.org/10.1007/BF02837825

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