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

Let p be a prime number and f∈Snl(Γ0(N),ψ) 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 = [4ordp(α0(p))] + 1 explained in the Main Theorem with the following properties: (i)For all pairs (s,χ) such that χ∈Xptors is a non-trivial Dirichlet characters, s∈ ℤ with 1 ≤ s ≤ l − n, s ≡ δ mod 2 and for s = 1 the character χ2 is non-trivial, the following equality holds(Formula presented.) where f0 is a modular form, associated to f and an embedding ip: ℚ¯ ↪ ℂp is fixed.(ii)For all pairs (s,χ) such that χ∈Xptors is a non-trivial Dirichlet character, s∈ ℤ with 1 − l + n ≤ s ≤ 0, s ≡ δ + 1 mod 2 the following equality holds(Formula presented.) Here, δ = 0 or 1 according to whether χ(−1) = 1 or χ(−1) = −1 and Λ∞(s), A(χ), Ep(s,ψ) are certain elementary factors including Gauss sum, Satake p-parameters, conductor cχ of Dirichlet character χ etc. © 2017, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.