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 hadmissible measures μ ^{+} and μ ^{−} for certain h = [4ord_{ p }(α _{0}(p))] + 1 explained in the Main Theorem with the following properties:

(i)
For all pairs (s,χ) such that \(\chi \in X_{p}^{\text {tors}}\) is a nontrivial Dirichlet characters, \(s\in \mathbb {Z}\) with 1 ≤ s ≤ l − n, s ≡ δ mod 2 and for s = 1 the character χ ^{2} is nontrivial, 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.

(ii)
For all pairs (s,χ) such that \(\chi \in X_{p}^{\text {tors}}\) is a nontrivial Dirichlet character, \(s\in \mathbb {Z}\) with 1 − l + n ≤ s ≤ 0, s ≡ δ + 1 mod 2 the following equality holds
$$\begin{array}{@{}rcl@{}} {\int}_{\mathbb{Z}_{p}^{\times}}\chi x_{p}^{s1}d\mu^{}&=&i_{p}\bigg(c_{\chi}^{n(1s)}A^{+}(\chi)\cdot E_{p}^{}(1s, \chi\chi^{0})\frac{{\Lambda}_{\infty}^{}(s)}{\langle f_{0}, f_{0}\rangle}\\ &&\times D^{(Np)}(f, 1s, \overline{\chi\chi^{0}})\bigg). \end{array} $$
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 pparameters, conductor c _{ χ } of Dirichlet character χ etc.
This is a preview of subscription content, access via your institution.
References
 1.
Amice, Y., Vëlu, J.: Distributions padiques associëes aux Sëries de Hecke. Astërisque 2425, 119–131 (1975)
 2.
Böcherer, S.: Über die Funktionalgleichung automorpher LFunktionen zur Siegelschen modulgrupe. J. Reine Angew. Math. 362, 146–168 (1985)
 3.
Böcherer, S., Schmidt, C.G.: padic measures attached to Siegel modular forms. Ann. Inst. Fourier 50, 1375–1443 (2000)
 4.
Courtieu, M., Panchishkin, A.: NonArchimedean LFunctions and Arithmetical Siegel Modular Forms. Lecture Notes in Mathematics, vol. 1471. SpringerVerlag, Berlin (2000)
 5.
Freitag, E.: Siegelsche Modulfunktionen. SpringerVerlag, Berlin (1983)
 6.
Ha, H.K.: padic interpolation and MellinMazur transform. Acta Math. Vietnam. 5, 77–99 (1981)
 7.
Panchishkin, A.A.: NonArchimedean LFunctions of Siegel and Hilbert Modular Forms. Lecture Notes in Mathematics, vol. 1471. SpringerVerlag, Berlin–Heidelberg (1991)
 8.
Vis̆ik, M. M.: NonArchimedean measures connected with Dirichlet series. Mat. Sb. 99, 248–260 (1976)
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 Grenoble1, 2014. The author would like to thank the anonymous referees of this manuscript whose remarks and corrections improved significantly the exposition.
Author information
Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Do, A.T. pAdic Admissible Measures Attached to Siegel Modular Forms of Arbitrary Genus. Vietnam J. Math. 45, 695–711 (2017). https://doi.org/10.1007/s100130170247x
Received:
Accepted:
Published:
Issue Date:
Keywords
 Siegel modular forms
 Special values
 Critical points
 Petersson product
 Rankin–Selberg method
 pAdic Lfunctions
Mathematics Subject Classification (2010)
 11F67
 11F46