scholarly journals Spectral decomposition of compactly supported Poincaré series and existence of cusp forms

2010 ◽  
Vol 146 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Goran Muić
Keyword(s):  
Spectral Decomposition ◽  
Poincaré Series ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Poincare Series ◽  
Compactly Supported ◽  
Classic Method

AbstractIn this paper we address the issue of existence of cusp forms by using an extension and refinement of a classic method involving (adelic) compactly supported Poincaré series. As a consequence of our adelic approach, we also deal with cusp forms for congruence subgroups.

ON THE NON-VANISHING OF CERTAIN MODULAR FORMS

2011 ◽  
Vol 07 (02) ◽  
pp. 351-370 ◽  
Author(s):  
GORAN MUIĆ
Keyword(s):  
Modular Forms ◽  
Fuchsian Group ◽  
Locally Compact Groups ◽  
Cusp Forms ◽  
Compact Groups ◽  
Congruence Subgroups ◽  
Locally Compact ◽  
Sharp Bounds ◽  
Poincare Series ◽  
Spanning Set

Let Γ ⊂ SL2(ℝ) be a Fuchsian group of the first kind. In this paper, we study the non-vanishing of the spanning set for the space of cuspidal modular forms [Formula: see text] of weight m ≥ 3 constructed in [5, Corollary 2.6.11]. Our approach is based on the generalization of the non-vanishing criterion for L1-Poincaré series defined for locally compact groups and proved in [6, Theorem 4.1]. We obtain very sharp bounds for the non-vanishing of the spaces of cusp forms [Formula: see text] for general Γ having at least one cusp. We obtain explicit results for congruence subgroups Γ(N), Γ0(N), and Γ1(N) (N ≥ 1).

scholarly journals Integral moments of automorphic L-functions

2008 ◽  
Vol 8 (2) ◽  
pp. 335-382 ◽  
Author(s):  
Adrian Diaconu ◽  
Paul Garrett
Keyword(s):  
Representation Theory ◽  
Number Field ◽  
Spectral Decomposition ◽  
Arbitrary Number ◽  
Number Fields ◽  
Rational Numbers ◽  
Poincaré Series ◽  
Poincare Series ◽  
Global And Local ◽  
Adele Group

AbstractWe obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.

Cusp Forms and Poincare Series

1968 ◽  
Vol 90 (2) ◽  
pp. 356 ◽  
Author(s):  
David Drasin
Keyword(s):  
Poincaré Series ◽  
Cusp Forms ◽  
Poincare Series

scholarly journals Automorphic pseudodifferential operators, Poincaré series and Eisenstein series

1999 ◽  
Vol 59 (1) ◽  
pp. 45-52
Author(s):  
Min Ho Lee
Keyword(s):  
Eisenstein Series ◽  
Pseudodifferential Operators ◽  
Poincaré Series ◽  
Cusp Forms ◽  
Explicit Formulas ◽  
Poincare Series

We construct Poincaré series and Eisenstein series for automorphic pseudodifferential operators, and show that the space of automorphic pseudodifferential operators associated to cusp forms is generated by Poincaré series. We also obtain explicit formulas for such Poincaré series and Eisenstein series.

scholarly journals Poincaré series, 3d gravity and averages of rational CFT

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Viraj Meruliya ◽  
Sunil Mukhi ◽  
Palash Singh
Keyword(s):  
Poincaré Series ◽  
Simons Theory ◽  
Partition Functions ◽  
Moody Algebra ◽  
Modular Invariant ◽  
Poincare Series ◽  
Single Genus ◽  
3D Gravity ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

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