### A certain Dirichlet series attached to Siegel modular forms of degree two.

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Let 𝓐₂(n) = Γ₂(n)∖𝔖₂ be the quotient of Siegel's space of degree 2 by the principal congruence subgroup of level n in Sp(4,ℤ). This is the moduli space of principally polarized abelian surfaces with a level n structure. Let 𝓐₂(n)* denote the Igusa compactification of this space, and ∂𝓐₂(n)* = 𝓐₂(n)* - 𝓐₂(n) its "boundary". This is a divisor with normal crossings. The main result of this paper is the determination of H(∂𝓐₂(n)*) as a module over the finite group Γ₂(1)/Γ₂(n). As an application...

Vector-valued Siegel modular forms may be found in certain cohomology groups with coefficients lying in an irreducible representation of the symplectic group. Using functoriality in the coefficients, we show that the ordinary components of the cohomology are independent of the weight parameter. The meaning of ordinary depends on a choice of parabolic subgroup of $GSp\left(4\right)$, giving a particular direction in the change of weight. Our results complement those of Taylor and Tilouine-Urban for the two other possible...

We employ recent results on Jacobi forms to investigate congruences and filtrations of Siegel modular forms of degree $2$. In particular, we determine when an analog of Atkin’s $U\left(p\right)$-operator applied to a Siegel modular form of degree $2$ is nonzero modulo a prime $p$. Furthermore, we discuss explicit examples to illustrate our results.