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We prove the meromorphic continuation and the functional equation of a twisted real-analytic Hermitain Eisenstein series of Klingen type, and as a consequence, deduce similar properties for the twisted Dirichlet series associated to a pair of Hermitian modular forms involving their Fourier–Jacobi coefficients. As an application of our result, we prove that infinitely many of the Fourier–Jacobi coefficients of a non-zero Hermitian cusp form do not vanish in any non-trivial arithmetic progression.

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $$\chi _{5}$$ , the cusp form of lowest weight for the group $${\text {Sp}}(2,\mathbb {Z})$$ . In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice $$A_{1}$$ and Igusa’s form $$\chi _{5}$$ appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates.

The mod p kernel of the theta operator is the set of modular forms whose image of the theta operator is congruent to zero modulo a prime p. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa’s odd weight cusp form is an element of mod 23 kernel of the theta operator. In this paper, we give some examples which represent elements in the mod p kernel of the theta operator in the case of Hermitian modular forms of degree 2.

In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\frak{p}$ in any ring of algebraic integers, mod $\frak{p}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan's congruence. We will conclude by giving numerical examples.

We discuss the Fourier–Jacobi expansion of certain vector valued Eisenstein series of degree $$2$$ , which is also real analytic. We show that its coefficients of index $$\pm 1$$ can be described by using a generating series of real analytic Jacobi forms. We also describe all the coefficients of general indices in suitable manners. Our method can be applied to study another Fourier series of Saito-Kurokawa type that is associated with a cusp form of one variable and half-integral weight. Then, following the arguments in the holomorphic case, we find that the Fourier series indeed defines a real analytic Siegel modular form of degree 2.

Under the hypothesis of the Resnikoff-Saldana conjecture, we give a precise formula for the abscissa of convergence of a series of Koecher-Maass type attached to a Siegel cusp form of arbitrary genus.

We generalize the classical Voronoi formula for $$r_{l}(n) = \#\{ (n_{1}, \ldots , n_{l}) \in \mathbf{Z}^{l}, n_{1}^{2} + \cdots + n_{l}^{2} = n \},$$ and as an application, we derive a sharp bound for the shifted convolution sum convolving the Fourier coefficients of holomorphic cusp forms with those of theta series.

We investigate degree two Siegel cusp forms of small weight for Γ0(p). Using the Restriction Technique we compute some dimensions and verify the conjectures ofHashimoto in some examples of weights three and four. For weight two we determine the dimension for primesp ≤ 41 and find only lifts. We explain in general how to compute spaces of Siegel cusp forms for subgroups of finite index in Γ n .

For a Siegel modular cusp formf of weightk letv(f) be the closure of the convex ray hull of the support of the Fourier series inside the cone of semidefinite forms. We show the existence of the extreme core,C ext, which satisfiesv(f) ⊇k Cext for all cusp forms. This is a generalization of the Valence Inequality to Siegel modular cusp forms. We give estimations of the extreme core for general n. For n ≤5 we use noble forms to improve these estimates. Forn = 2 we almost specify the extreme core but fall short. We supply improved estimates for all relevant constants and show optimality in some cases. The techniques are mainly from the geometry of numbers but we also use IGUSA’s generators for the ring of Siegel modular forms in degree two.

To any finite symmetric subsetR ⊂ O3 corresponds a Hecke operatorTR on L2(S2) which leaves the eigenspaces ℋn (n ≥ 0) of the Laplacian invariant. We compute the trace ofTR | ℋn and prove that the sum of the positive eigenvalues ofTR on ⊕k=0n-1ℋk prevails over the modulus of the sum of the negative eigenvalues. For anym ∈ ℕ the integral quaternions of normm define such a Hecke operator\(T_{R_m } \), and renormalizing the traces ofTRm | ℋn slightly, we obtain sequences of Fourier coefficients of modular forms on Γ0(4).

A finite number ofL-functions are associated to every Jacobi cusp form of degreen. TheseL-functions are infinite series constructed with the Fourier coefficients of the form and a variables in ℂn. It is proved that eachL-function has an integral representation, admits a holomorphic continuation to the whole space ℂn, and the row vector formed with them satisfies a particular matrix functional equation.

Every Siegel modular form has a Fourier-Jacobi expansion. This paper provides various sets of Fourier coefficients whose vanishing implies that the associated cusp form is identically zero. We call such setsestimates because in the Fourier series case, an upper bound for the dimension of the vector space of cusp forms is provided by the cardinality of the set. Our general estimates have, among others, those estimates of Siegel and Eichler as corollaries. In particular, one new corollary of our general estimates appears to be superior for computational purposes to all other known estimates. To illustrate the use of this corollary, we prove the known result that the theta series of the latticesD 16 + andE 8 ⊕E 8 are the same in degreen = 3 by computing just one Fourier coefficient. 1991 Mathematics Subject Classification. 11F46 (11F30, 11F27).