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On some Poincaré-series on hyperbolic space

On some Poincaré-series on hyperbolic space Forum Math. 11 (1999), 483±502 ( de Gruyter 1999  On some Poincare-series on hyperbolic space Roland Matthes (Communicated by Peter Sarnak)   Abstract. We relate theta-lifts of real-analytic Poincare-series to hyperbolic distance Poincareseries on n-dimensional hyperbolic space averaged over Heegner points and hyperplanes, respectively. As a consequence we can prove at least for the three dimensional case a generalization of a formula of Maass for the Fourier-coe½cients of theta-lifted cusp forms. 1991 Mathematics Subject Classi®cation: 11F. 1 Introduction Let H n denote hyperbolic n-space with dPY Q the hyperbolic distance between the points P and Q. Introduce dPY Q 2 coshdPY Q then it is easily seen (see e.g. [5]) that dPY Q nÀ2 i0 ui P À ui Q 2 v 2 P v 2 Q vPvQ where u0 Y F F F Y unÀ2 Y v are the coordinates in the upper half-space model, see section 2. From the de®nition follows that d is a point pair invariant, i.e. for g from the isometry group of H n we have dgPY Q dPY gÀ1 QX Let q be a discrete co®nite subgroup of Iso H n , of orientation preserving  isometries of H http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

On some Poincaré-series on hyperbolic space

Forum Mathematicum , Volume 11 (4) – Jun 1, 1999

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Publisher
de Gruyter
Copyright
Copyright (c)1999 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1999.010
Publisher site
See Article on Publisher Site

Abstract

Forum Math. 11 (1999), 483±502 ( de Gruyter 1999  On some Poincare-series on hyperbolic space Roland Matthes (Communicated by Peter Sarnak)   Abstract. We relate theta-lifts of real-analytic Poincare-series to hyperbolic distance Poincareseries on n-dimensional hyperbolic space averaged over Heegner points and hyperplanes, respectively. As a consequence we can prove at least for the three dimensional case a generalization of a formula of Maass for the Fourier-coe½cients of theta-lifted cusp forms. 1991 Mathematics Subject Classi®cation: 11F. 1 Introduction Let H n denote hyperbolic n-space with dPY Q the hyperbolic distance between the points P and Q. Introduce dPY Q 2 coshdPY Q then it is easily seen (see e.g. [5]) that dPY Q nÀ2 i0 ui P À ui Q 2 v 2 P v 2 Q vPvQ where u0 Y F F F Y unÀ2 Y v are the coordinates in the upper half-space model, see section 2. From the de®nition follows that d is a point pair invariant, i.e. for g from the isometry group of H n we have dgPY Q dPY gÀ1 QX Let q be a discrete co®nite subgroup of Iso H n , of orientation preserving  isometries of H

Journal

Forum Mathematicumde Gruyter

Published: Jun 1, 1999

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