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Integral moments of automorphic L-functions

Part of: Algebraic number theory: global fields Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  16 October 2008

Adrian Diaconu  and
Paul Garrett
Adrian Diaconu
Affiliation:
School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA (cad@math.umn.edu)
Paul Garrett
Affiliation:
School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, MN 55455, USA (garrett@math.umn.edu)
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Abstract

We 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.

Keywords

integral momentsPoincaré seriesEisenstein seriesL-functionsspectral decompositionmeromorphic continuation

MSC classification

Secondary: 11R42: Zeta functions and $L$-functions of number fields 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11M41: Other Dirichlet series and zeta functions 11R47: Other analytic theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 2 , April 2009 , pp. 335 - 382
Copyright
Copyright © Cambridge University Press 2009

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