Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T14:02:53.274Z Has data issue: false hasContentIssue false
  • English
  • Français

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, ibukiyam@math.sci.osaka-u.ac.jp
Hiroshi Saito
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Osaka 560-0043, Japan, ibukiyam@math.sci.osaka-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1] Christian, U., Berechnung des Ranges der Schar der Spitzenformen zur Modulgruppe zweiten Grades und Stufe q > 2, J. Reine Angew. Math. 277 (1975), 130–154.Google Scholar
[2] Christian, U., Zur Berechnung des Ranges der Schar der Spitzenformen zur Modulgruppe zweiten Grades und Stufe q > 2, J. Reine Angew. Math. 296 (1977), 108–118.Google Scholar
[3] Cohen, H., Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), 271–285.Google Scholar
[4] Goldfeld, D. and Hoffstein, J., Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series, Invent. Math. 80 (1985), 185–208.Google Scholar
[5] Ibukiyama, T. and Katsurada, H., “Koecher-Maass series for real analytic Siegel Eisenstein series” in Automorphic Forms and Zeta Functions (Tokyo, 2004), World Sci., Hackensack, NJ, 2006, 170–197.CrossRefGoogle Scholar
[6] Ibukiyama, T. and Saito, H., On zeta functions associated to symmetric matrices and an explicit conjecture on dimension of Siegel modular forms of general degree, Int. Math. Res. Not. IMRN 1992, no. 8, 161–169.Google Scholar
[7] Ibukiyama, T. and Saito, H., On zeta functions associated to symmetric matrices, I: An explicit form of zeta functions, Amer. J. Math. 117 (1995), 1097–1155.Google Scholar
[8] Ibukiyama, T. and Saito, H., On zeta functions associated to symmetric matrices, III: An explicit form of L-functions, Nagoya Math. J. 146 (1997), 149–183.Google Scholar
[9] Ibukiyama, T. and Saito, H., On “easy” zeta functions [translation of Sūgaku 50 (1998), 1–11], Sugaku Expositions 14 (2001), 191–204.Google Scholar
[10] Kohnen, W., Modular forms of half-integral weight on Γ0(4), Math. Ann. 248 (1980), 249–266.Google Scholar
[11] Morita, Y., An explicit formula for the dimension of spaces of Siegel modular forms of degree two, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 167–248.Google Scholar
[12] Saito, H., Explicit form of the zeta functions of prehomogeneous vector spaces, Math. Ann. 315 (1999), 587–615.Google Scholar
[13] Satake, I., “On zeta functions associated with self-dual homogeneous cones” in Number Theory and Related Topics (Bombay, 1988), Tata Inst. Fund. Res. Stud. Math. 12, Tata Inst. Fund. Res., Bombay, 1989, 177–193.Google Scholar
[14] Satake, I. and Faraut, J., The functional equation of zeta distributions associated with formally real Jordan algebras, Tohoku Math. J. (2) 36 (1984), 469–482.Google Scholar
[15] Satake, I. and Ogata, S., “Zeta functions associated to cones and their special values” in Automorphic Forms and Geometry of Arithmetic Varieties, Adv. Stud. Pure Math. 15, Academic Press, Boston, 1989, 1–27.Google Scholar
[16] Satō, F., On zeta functions of ternary zero forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 585–604.Google Scholar
[17] Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170.Google Scholar
[18] Shimura, G., “Modular forms of half integral weight” in Modular Forms of One Variable, I (Antwerp, 1972), Lecture Notes in Math. 320, Springer, Berlin, 1973, 57–74.Google Scholar
[19] Shimura, G., On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481.Google Scholar
[20] Shimura, G., On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc. (3) 31 (1975), 79–98.Google Scholar
[21] Shintani, T., On zeta functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 25–65.Google Scholar
[22] Siegel, C. L., Uber die Zetafunktionen indefiniter quadratischer Formen, Math. Z. 43 (1938), 682–708.Google Scholar
[23] Sturm, J., Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980), 219–240.Google Scholar
[24] Tsushima, R., A formula for the dimension of spaces of Siegel cusp forms of degree three, Amer. J. Math. 102 (1980), 937–977.Google Scholar
[25] Yamazaki, T., On Siegel modular forms of degree two, Amer. J. Math. 98 (1976), 39–53.CrossRefGoogle Scholar
[26] Yukie, A., Shintani Zeta Functions, London Math. Soc. Lecture Note Ser. 183, Cambridge University Press, Cambridge, 1993.Google Scholar
[27] Zagier, D. B., “Modular forms whose coefficents involve zeta functions of quadratic fields” in Modular Forms of One Variable, VI (Bonn, 1976), Lecture Notes in Math. 627, Springer, Berlin, 1977, 105–169.Google Scholar