Poincaré series for SO(n, 1)

Jian Shu Li; I. Piatetski-Shapiro; P. Sarnak

doi:10.1007/BF02837825

Poincaré series for SO(n, 1)

Jian Shu Li, I. Piatetski-Shapiro, P. Sarnak

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

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.

Original language	English (US)
Pages (from-to)	231-237
Number of pages	7
Journal	Proceedings of the Indian Academy of Sciences - Mathematical Sciences
Volume	97
Issue number	1-3
DOIs	https://doi.org/10.1007/BF02837825
State	Published - Dec 1987
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

Keywords

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

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10.1007/BF02837825

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title = "Poincar{\'e} series for SO(n, 1)",

abstract = "A theory of Poincar{\'e} 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 = "Lobachevsky space, Poincar{\'e} series, Selberg-Kloosterman zeta function, non-uniform lattices",

author = "Li, {Jian Shu} and I. Piatetski-Shapiro and P. Sarnak",

year = "1987",

month = dec,

doi = "10.1007/BF02837825",

language = "English (US)",

volume = "97",

pages = "231--237",

journal = "Proceedings of the Indian Academy of Sciences - Mathematical Sciences",

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TY - JOUR

T1 - Poincaré series for SO(n, 1)

AU - Li, Jian Shu

AU - Piatetski-Shapiro, I.

AU - Sarnak, P.

PY - 1987/12

Y1 - 1987/12

N2 - 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.

AB - 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.

KW - Lobachevsky space

KW - Poincaré series

KW - Selberg-Kloosterman zeta function

KW - non-uniform lattices

UR - http://www.scopus.com/inward/record.url?scp=51249175741&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249175741&partnerID=8YFLogxK

U2 - 10.1007/BF02837825

DO - 10.1007/BF02837825

M3 - Article

AN - SCOPUS:51249175741

SN - 0370-0089

VL - 97

SP - 231

EP - 237

JO - Proceedings of the Indian Academy of Sciences - Mathematical Sciences

JF - Proceedings of the Indian Academy of Sciences - Mathematical Sciences

IS - 1-3

ER -

Poincaré series for SO(n, 1)

Abstract

All Science Journal Classification (ASJC) codes

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