Journal of the Mathematical Society of Japan

An explicit dimension formula for Siegel cusp forms with respect to the non-split symplectic groups

Hidetaka KITAYAMA

Full-text: Open access

Abstract

We give an explicit dimension formula for the spaces of vector valued Siegel cusp forms of degree two with respect to a certain kind of arithmetic subgroups of the non-split Q-forms of Sp(2,R).

Article information

Source
J. Math. Soc. Japan, Volume 63, Number 4 (2011), 1263-1310.

Dates
First available in Project Euclid: 27 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1319721141

Digital Object Identifier
doi:10.2969/jmsj/06341263

Mathematical Reviews number (MathSciNet)
MR2855813

Zentralblatt MATH identifier
1267.11054

Subjects
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F72: Spectral theory; Selberg trace formula

Keywords
Siegel modular forms dimension formula non-split Q-forms

Citation

KITAYAMA, Hidetaka. An explicit dimension formula for Siegel cusp forms with respect to the non-split symplectic groups. J. Math. Soc. Japan 63 (2011), no. 4, 1263--1310. doi:10.2969/jmsj/06341263. https://projecteuclid.org/euclid.jmsj/1319721141


Export citation

References

  • H. Aoki and T. Ibukiyama, Simple graded rings of Siegel modular forms, differential operators and Borcherds products, Internat. J. Math., 16 (2005), 249–279.
  • T. Arakawa, Automorphic forms on quaternion unitary group of degree 2 (in Japanese), Master thesis, University of Tokyo, 1975.
  • T. Arakawa, The dimension of the space of cusp forms on the Siegel upper half plane of degree two related to a quaternion unitary group, J. Math. Soc. Japan, 33 (1981), 125–145.
  • K. Hashimoto, On Brandt matrices associated with the positive definite quaternion Hermitian forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 227–245.
  • K. Hashimoto, The dimension of the spaces of cusp forms on Siegel upper half plane of degree two I, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1983), 403–488.
  • K. Hashimoto, The dimension of the spaces of cusp forms on Siegel upper half plane of degree two II, The $\mathbf{Q}$-rank one case, Math. Ann., 266 (1984), 539–559.
  • K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 549–601.
  • K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms (II), J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1982), 695–699.
  • K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms (III), J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1983), 393–401.
  • K. Hashimoto and T. Ibukiyama, On relations of dimensions of automorphic forms of $Sp(2,\bR)$ and its compact twist $Sp(2)$ (II), Automorphic forms and number theory, Adv. Stud. Pure Math., 7 (1985), 31–102.
  • Y. Hirai, On Eisenstein series on quaternion unitary groups of degree 2, J. Math. Soc. Japan, 51 (1999), 93–128.
  • T. Ibukiyama, On symplectic Euler factors of genus two, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 587–614.
  • T. Ibukiyama, On relations of dimensions of automorphic forms of $Sp(2,\mathbf{R})$ and its compact twist $Sp(2)$ (I), Automorphic forms and number theory, Adv. Stud. Pure Math., 7 (1985), 7–30.
  • T. Ibukiyama, On differential operators on automorphic forms and invariant pluri-harmonic polynomials, Comment. Math. Univ. St. Pauli, 48 (1999), 103–118.
  • T. Ibukiyama, Paramodular forms and compact twist, In: Automorphic Forms on $GSp(4)$, Proceedings of the 9th Autumn Workshop on Number Theory, (ed. M.Furusawa), 2007, pp.,37–48.
  • T. Ibukiyama, Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces, Proceedings of the 4-th Spring Conference, 2007, pp.,39–60.
  • R. P. Langlands, The dimension of spaces of automorphic forms, Amer. J. Math., 85 (1963), 99–125.
  • Y. Morita, 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.
  • V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Academic Press, 1994.
  • G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15 (1963), 33–65.
  • S. Wakatsuki, Dimension formulas for spaces of vector valued Siegel cusp forms of degree two, to appear in J. Number Theory.