Nagoya Mathematical Journal

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

Tomoyoshi Ibukiyama and Hiroshi Saito

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Abstract

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.

Article information

Source
Nagoya Math. J., Volume 208 (2012), 265-316.

Dates
First available in Project Euclid: 5 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1354716562

Digital Object Identifier
doi:10.1215/00277630-1815258

Mathematical Reviews number (MathSciNet)
MR3006702

Zentralblatt MATH identifier
1275.11085

Subjects
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11S90: Prehomogeneous vector spaces
Secondary: 11E12: Quadratic forms over global rings and fields 11R99: None of the above, but in this section

Citation

Ibukiyama, Tomoyoshi; Saito, Hiroshi. On zeta functions associated to symmetric matrices, II: Functional equations and special values. Nagoya Math. J. 208 (2012), 265--316. doi:10.1215/00277630-1815258. https://projecteuclid.org/euclid.nmj/1354716562


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