Skip to main content

p-Adic Admissible Measures Attached to Siegel Modular Forms of Arbitrary Genus

Abstract

Let p be a prime number and \(f\in {S_{n}^{l}}({\Gamma }_{0}(N), \psi )\) be a Siegel cusp eigenform of genus n. We consider the standard zeta function D (Np)(f,s,χ), which takes algebraic values at critical points after normalization. We construct two h-admissible measures μ + and μ for certain h = [4ord p (α 0(p))] + 1 explained in the Main Theorem with the following properties:

  1. (i)

    For all pairs (s,χ) such that \(\chi \in X_{p}^{\text {tors}}\) is a non-trivial Dirichlet characters, \(s\in \mathbb {Z}\) with 1 ≤ sln, sδ mod 2 and for s = 1 the character χ 2 is non-trivial, the following equality holds

    $${\int}_{\mathbb{Z}_{p}^{\times}}\chi x_{p}^{-s}d\mu^{+}=i_{p}\left( c_{\chi}^{s(n+1)}A^{+}(\chi)\cdot E_{p}^{+}(s, \chi\chi^{0})\frac{{\Lambda}_{\infty}^{+}(s)}{\langle f_{0}, f_{0}\rangle}\cdot D^{(Np)}(f, s, \overline{\chi\chi^{0}})\right), $$

    where f 0 is a modular form, associated to f and an embedding \(i_{p}:\overline {\mathbb {Q}}\hookrightarrow \mathbb {C}_{p}\) is fixed.

  2. (ii)

    For all pairs (s,χ) such that \(\chi \in X_{p}^{\text {tors}}\) is a non-trivial Dirichlet character, \(s\in \mathbb {Z}\) with 1 − l + ns ≤ 0, sδ + 1 mod 2 the following equality holds

    $$\begin{array}{@{}rcl@{}} {\int}_{\mathbb{Z}_{p}^{\times}}\chi x_{p}^{s-1}d\mu^{-}&=&i_{p}\bigg(c_{\chi}^{n(1-s)}A^{+}(\chi)\cdot E_{p}^{-}(1-s, \chi\chi^{0})\frac{{\Lambda}_{\infty}^{-}(s)}{\langle f_{0}, f_{0}\rangle}\\ &&\times D^{(Np)}(f, 1-s, \overline{\chi\chi^{0}})\bigg). \end{array} $$

Here, δ = 0 or 1 according to whether χ(−1) = 1 or χ(−1) = −1 and Λ (s), A(χ), E p (s,ψ) are certain elementary factors including Gauss sum, Satake p-parameters, conductor c χ of Dirichlet character χ etc.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Amice, Y., Vëlu, J.: Distributions p-adiques associëes aux Sëries de Hecke. Astërisque 24-25, 119–131 (1975)

    MATH  Google Scholar 

  2. 2.

    Böcherer, S.: Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen modulgrupe. J. Reine Angew. Math. 362, 146–168 (1985)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Böcherer, S., Schmidt, C.-G.: p-adic measures attached to Siegel modular forms. Ann. Inst. Fourier 50, 1375–1443 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Courtieu, M., Panchishkin, A.: Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms. Lecture Notes in Mathematics, vol. 1471. Springer-Verlag, Berlin (2000)

  5. 5.

    Freitag, E.: Siegelsche Modulfunktionen. Springer-Verlag, Berlin (1983)

    Book  MATH  Google Scholar 

  6. 6.

    Ha, H.K.: p-adic interpolation and Mellin-Mazur transform. Acta Math. Vietnam. 5, 77–99 (1981)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Panchishkin, A.A.: Non-Archimedean L-Functions of Siegel and Hilbert Modular Forms. Lecture Notes in Mathematics, vol. 1471. Springer-Verlag, Berlin–Heidelberg (1991)

  8. 8.

    Vis̆ik, M. M.: Non-Archimedean measures connected with Dirichlet series. Mat. Sb. 99, 248–260 (1976)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank his adviser, Professor Alexei Panchishkin, for all assistance during the 3 years of research at the Institute Fourier. This paper is based on the author’s PhD thesis in the University Grenoble-1, 2014. The author would like to thank the anonymous referees of this manuscript whose remarks and corrections improved significantly the exposition.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anh Tuan Do.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Do, A.T. p-Adic Admissible Measures Attached to Siegel Modular Forms of Arbitrary Genus. Vietnam J. Math. 45, 695–711 (2017). https://doi.org/10.1007/s10013-017-0247-x

Download citation

Keywords

  • Siegel modular forms
  • Special values
  • Critical points
  • Petersson product
  • Rankin–Selberg method
  • p-Adic L-functions

Mathematics Subject Classification (2010)

  • 11F67
  • 11F46