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On zeta functions associated to symmetric matrices, II: Functional equations and special values

Published online by Cambridge University Press:  11 January 2016

Tomoyoshi Ibukiyama  and
Hiroshi Saito
Tomoyoshi Ibukiyama
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Osaka 560-0043, Japan, [email protected]
Hiroshi Saito
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Osaka 560-0043, Japan, [email protected]
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Abstract

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New simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.

Keywords

11F4611S9011E1211R99
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 208: Memorial Volume for Professor Hiroshi Saito , December 2012 , pp. 265 - 316
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

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