Nagoya Mathematical Journal

Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

Takahiko Ueno

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Abstract

In this paper, we prove the functional equations for the zeta functions in two variables associated with prehomogeneous vector spaces acted on by maximal parabolic subgroups of orthogonal groups. Moreover, applying the converse theorem of Weil type, we show that elliptic modular forms of integral or half integral weight can be obtained from the zeta functions.

Article information

Source
Nagoya Math. J., Volume 175 (2004), 1-37.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114632092

Mathematical Reviews number (MathSciNet)
MR2085308

Zentralblatt MATH identifier
1075.11042

Subjects
Primary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11S90: Prehomogeneous vector spaces

Citation

Ueno, Takahiko. Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms. Nagoya Math. J. 175 (2004), 1--37. https://projecteuclid.org/euclid.nmj/1114632092


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References

  • H. Cohen, Sums involving the values at negative integers of $L$-functions of quadratic characters , Math. Ann., 217 (1975), 271–285.
  • R. Hörmander, An introduction to complex analysis in several variables, North- Holland (1973).
  • T. Miyake, Modular forms, Springer (1989).
  • M. Muro, A note on the holonomic system of invariant hyperfunctions on a certain prehomogeneous vector space , Algebraic Analysis, vol. 2 (M. Kashiwara and T. Kawai, eds.), Academic Press (1988), 493–503.
  • M. Peter, Dirichlet series in two variables , J. reine angew. Math., 522 (2000), 27–50.
  • F. Sato, Zeta functions in several variables associated with prehomogeneous vector spaces I: functional equations , Tohoku Math. J., 34 (1982), 437–483.
  • F. Sato, Zeta functions in several variables associated with prehomogeneous vector spaces II: a convergence criterion , Tohoku Math. J., 35 (1983), 77–99.
  • F. Sato, On functional equations of zeta distributions , Adv. Studies in pure Math., 15 (1989), 465–508.
  • F. Sato, $L$-functions of prehomogeneous vector spaces , preprint (2003).
  • G. Shimura, On modular forms of half integral weight , Ann. of Math., 97 (1973), 440–481.
  • T. Shintani, On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms , J. Math. Soc. Japan, 24 (1972), 132–188.
  • T. Shintani, On zeta-functions associated with the vector space of quadratic forms , J. Fac. Sci. Univ. Tokyo, Sect. IA 22 (1975), 25–65.
  • H. M. Stark, $L$-functions and character sums for quadratic forms (I) , Acta Arithmetica, XIV (1968), 35–50.
  • T. Ueno, Elliptic modular forms arising from zeta functions in two-variable attached to the space of binary hermitian forms , J. Number Theory, 86 (2001), 302–329.
  • A. Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , Math. Ann., 168 (1967), 149–156.