Nagoya Mathematical Journal
- Nagoya Math. J.
- Volume 208 (2012), 265-316.
On zeta functions associated to symmetric matrices, II: Functional equations and special values
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.
Nagoya Math. J., Volume 208 (2012), 265-316.
First available in Project Euclid: 5 December 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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