The functional equation for the twisted spinor L-series of genus 2



We prove the functional equation for the twisted spinor L-series of a cuspidal, holomorphic Siegel eigenform for the full modular group of genus 2. It follows from a more general functional equation, valid for Rankin convolutions of paramodular cuspforms. A non-vanishing result for Fourier-Jacabi coefficients of the eigenforms in question is the central pillar of the deduction of the former from the latter functional equation.


Siegel modular forms Paramodular group Spinor zeta-function Fourier-Jacobi expansion Rankin convolution 

Mathematics Subject Classification

11F46 11F50 11F66 


  1. 1.
    Andrianov, A.N.: Euler products that correspond to Siegel’s modular forms of genus 2. Usp. Mat. Nauk 29(3), 43–110 (1974) MathSciNetMATHGoogle Scholar
  2. 2.
    Andrianov, A.N.: L-functions of Siegel modular forms and twist operators. MPI-Preprint 4152 (2010) Google Scholar
  3. 3.
    Bump, D.: Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1997) CrossRefGoogle Scholar
  4. 4.
    Cogdell, J., Piatetski-Shapiro, I.: A converse theorem for GL4. Math. Res. Lett. 3(1), 67–76 (1996) MathSciNetMATHGoogle Scholar
  5. 5.
    Cogdell, J., Piatetski-Shapiro, I.: Converse theorems for \({\rm GL}_{n}\) II. J. Reine Angew. Math. 507, 165–188 (1999) MathSciNetMATHGoogle Scholar
  6. 6.
    Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Birkhäuser, Boston (1985) MATHGoogle Scholar
  7. 7.
    Freitag, E.: Siegelsche Modulfunktionen. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1983) MATHCrossRefGoogle Scholar
  8. 8.
    Gritsenko, V.A.: Arithmetical lifting and its applications. In: David, S. (ed.) Number Theory. Séminaire de Théorie des Nombres de Paris 1992–1993, pp. 103–126. Cambridge University Press, Cambridge (1995) Google Scholar
  9. 9.
    Gritsenko, V.A.: Modulformen zur Paramodulgruppe und Modulräume der abelschen Varietäten. Mathematica Gottingensis, vol. 12, pp. 1–89 (1995) Google Scholar
  10. 10.
    Kohnen, W., Krieg, A., Sengupta, J.: Characteristic twists of a Dirichlet series for Siegel cusp forms. Manuscr. Math. 87, 489–499 (1995) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Krieg, A.: Hecke algebras. Mem. Am. Math. Soc. 435, 158 (1990) MathSciNetGoogle Scholar
  12. 12.
    Kohnen, W., Skoruppa, N.: A certain Dirichlet series attached to Siegel modular forms of degree two. Invent. Math. 95, 541–558 (1989) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kuß, M.: Die Funktionalgleichung der getwisteten Spinorzetafunktion und die Böcherer Vermutung. Ph.D. Thesis, Ruprecht-Karls-Universität, Heidelberg (2002) Google Scholar
  14. 14.
    Matsuda, I.: Dirichlet series corresponding to Siegel modular forms of degree 2, level N. Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 28(1), 21–49 (1978) MathSciNetMATHGoogle Scholar
  15. 15.
    Neukrich, J.: Algebraische Zahlentheorie. Springer, Berlin (1992) Google Scholar
  16. 16.
    Pitale, A., Saha, A., Schmidt, R.: Transfer of Siegel cusp forms of degree 2 (2011). arXiv:1106.5611 [math.NT]
  17. 17.
    Rombach, A.: Über Twists in Siegelschen Modulformen zweiten Grades und ihre Spinorzetafunktion. Ph.D. Thesis, Universität Mannheim (2004) Google Scholar
  18. 18.
    Terras, A.: Harmonic analysis on symmetric spaces and applications, vol. I. Springer, New York (1985) MATHCrossRefGoogle Scholar
  19. 19.
    Zagier, D.: Zetafunktionen und quadratische Körper. Springer, Berlin (1981) MATHCrossRefGoogle Scholar

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Lehrstuhl A für MathematikRWTH Aachen UniversityAachenGermany
  2. 2.Dep. MathematikETH ZürichZürichSwitzerland

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