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We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight g 2 and genus g. When g = 4 , the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. In this paper we compute the Fourier coefficients of the Siegel Eisenstein series of degree 2, level p with the trivial or the quadratic character, associated to a certain cusp. For that we need to define the p-factor of the special type of Siegel series with character. [ABSTRACT FROM AUTHOR]

We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight. [ABSTRACT FROM AUTHOR]

We study analogues of Sturm's bounds for vector valued Siegel modular forms of type (k, 2), which was already studied by Sturm in the case of an elliptic modular form and by Choi–Choie–Kikuta, Poor–Yuen and Raum–Richter in the case of scalar valued Siegel modular forms. [ABSTRACT FROM AUTHOR]

We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform of half-integral weight. [ABSTRACT FROM AUTHOR]

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct χ5, the cusp form of lowest weight for the group Sp(2,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 A1 and Igusa's form χ5 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. [ABSTRACT FROM AUTHOR]

In this survey, we present recent results of the authors about non-meromorphic modular objects known as polar harmonic Maass forms. These include the computation of Fourier coefficients of meromorphic modular forms and relations between inner products of meromorphic modular forms and higher Green's functions evaluated at CM-points. [ABSTRACT FROM AUTHOR]

We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficients. The method in this paper is essentially a generalization of Kitaoka's previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We give a formula for the Fourier coefficients of the image of the Saito-Kurokawa lifting over a totally real number field $$K$$ , and we prove the image of the lifting satisfies a generalization of the Maass relation. We also give an explicit form of a Siegel series of degree 2 for any finite extension of $${\mathbb {Q}}_p$$ . [ABSTRACT FROM AUTHOR]