Related papers: Characterizations of the Saito-Kurokawa lifting: a…

We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in [Buz05], discuss strategies for making further progress, and examine other…

We prove an equidistribution statement for the Satake parameters of the local representations attached to Siegel cusp forms of degree $2$ of increasing level and weight, counted with a certain arithmetic weight. We then apply this to…

Jacobi-Forms can be decomposed as a linear combination of Thetafunctions with modular forms as coefficients. It is shown that the space of these coefficient modular forms of Fourier-Jacobi-Forms, which come from Siegel cusp forms, has full…

We review how some multianalytic inner functions of the Beurling type theorem are associated to row contractions following works of G.Popescu. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for…

We show that the meromorphic modular forms recently considered by Bringmann and Kane can be obtained as images of regularized theta lifts of Poincar\'{e} series under weight raising operators. We use this fact in order to simplify the…

Let $\mathfrak{g}$ be a simple complex Lie algebra of classical type with a Cartan subalgebra $\mathfrak{h}$. We fix a standard parabolic subalgebra $\mathfrak{p}\supset \mathfrak{h}$. The socular simple modules are just those highest…

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models,…

We prove an algebraicity property for a certain ratio of Petersson norms associated to a Siegel cusp form of degree 2 (and arbitrary level) whose adelization generates a weak endoscopic lift. As a preparation for this, we explicate various…

In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the…

Kaneko and Sakai recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and…

We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and…

Let $\frak T_2$ (resp. $\mathfrak{T}$) be the Hermitian symmetric domain of $Spin(2,10)$ (resp. $E_{7,3}$). In the previous work, we constructed holomorphic cusp forms on $\mathfrak{T}$ from elliptic cusp forms with respect to…

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We…

In this paper the image of the Saito-Kurokawa lift of level $N$ with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level $N$ by symmetries involving Hecke operators related to…

Let $\f$ be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution $L$-values $L(\f,\g,s)$, where $\g$ is a theta-lift modular form corresponding to a finite-order character. We prove weak forms…

This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties;…

We identify the spectral curve of pure gauge SU(2) Seiberg-Witten theory with the Weierstrass curve $\mathbbm{C}/L \ni z \mapsto (1,\wp(z),\wp(z)')$ and thereby obtain explicitely a modular form from which the moduli space parameter $u$ and…

The possibility of defining sesquilinear forms starting from one or two sequences of elements of a Hilbert space is investigated. One can associate operators to these forms and in particular look for conditions to apply representation…

In this paper, we prove a conjecture of Broadhurst and Zudilin \cite{BZ17} concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds…

This paper presents five new examples of modular rigid Calabi-Yau threefolds arising from the modular elliptic surface of level 6. Explicit correspondences to newforms of weight 4 and level 10, 17, 21, and 73 are given.