Poincaré series and the spectral decomposition of L2(Γ \H, χ)

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.

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

Compositio Mathematica ◽

10.1112/s0010437x09004473 ◽

2010 ◽

Vol 146 (1) ◽

pp. 1-20 ◽

Cited By ~ 6

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.

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Poincaré series, 3d gravity and averages of rational CFT

Journal of High Energy Physics ◽

10.1007/jhep04(2021)267 ◽

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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