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:

• (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 ≤ s ≤ l − n, s ≡ δ 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.

• (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 + n ≤ s ≤ 0, s ≡ δ + 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.