Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms
type: Item
instance of: scholarly article
title: Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms
zbMATH Open document ID: 0799.11012
DOI: 10.1007/BF02830877
author: Fumihiro Sato
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1994-11-20T00:00:00Z
review text: The purpose of this paper is to extend the range of zeta functions associated with prehomogeneous vector spaces. For an algebraic number field \(k\), we take a prehomogeneous vector space \((G_ k, V_ k)\) defined over \(k\). Roughly speaking, zeta functions associated with \((G_ k, V_ k)\) are obtained as a Dirichlet series whose coefficients are numbers of \(G_ k\)-orbits in \(V_ k\setminus S_ k\) (\(S_ k\) is a singular set of \((G_ k, V_ k)\)). However such definition excludes the cases related to the automorphic forms, for example Hecke's \(L\)-function. In this paper, the author tries to reconstruct the old general theory so that we can deal with the zeta functions in a wider class including them. The author generalizes the theory of prehomogeneous vector spaces with symmetric structure of \(K_ \varepsilon\)-type and the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character.
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 562806
zbMATH Keywords: range of zeta functions
zbMATH Keywords: prehomogeneous vector spaces
zbMATH Keywords: functional equation
zbMATH Keywords: automorphic forms
reviewed by: Masakazu Muro
MaRDI profile type: Publication
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cites work: Q3766269
cites work: Meromorphic property of the functions \(P^ \lambda\)
cites work: Q3480288
cites work: Density and maximality of arithmetic subgroups.
cites work: Arithmetic subgroups of algebraic groups
cites work: Q3049039
cites work: Q3481949
cites work: Zeta functions of simple algebras
cites work: Automorphic forms on semisimple Lie groups. Notes by J. G. M. Mars.
cites work: Some Dirichlet series with coefficients related to periods of automorphic eigenforms
cites work: Fundamental Solutions of Invariant Differential Operators on Symmetric Spaces
cites work: A duality for symmetric spaces with applications to group representations. II: Differential equations and eigenspace representations
cites work: Q3331943
cites work: Some Results on Invariant Differential Operators on Symmetric Spaces
cites work: Heegner points, cycles and Maass forms
cites work: On holonomic systems for \(\prod_{l=1}^N(f_l+\sqrt{-1}0)^{\lambda_l}\)
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cites work: Q3237357
cites work: Zeta-functions with Größencharakteren and spherical functions
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cites work: Siegel's modular forms and Dirichlet series. Course given at the University of Maryland, 1969-1970
cites work: Orbits on affine symmetric spaces under the action of parabolic subgroups
cites work: Poisson transformations on affine symmetric spaces
cites work: Eigenspaces of invariant differential operators on an affine symmetric space
cites work: Analysis on real hyperbolic spaces
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cites work: Q3769182
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cites work: Q3826640
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cites work: Q3137241
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description: scientific article; zbMATH DE number 562806