Related papers: Weak Maass-Poincare series and weight 3/2 mock mod…

Let $\rho$ denote an irreducible two-dimensional representation of $\Gamma_{0}(2)$. The collection of vector-valued modular forms for $\rho$, which we denote by $M(\rho)$, form a graded and free module of rank two over the ring of modular…

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this…

We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincar\'e series in a companion paper. The source term of the Laplace equation is a product of…

Given a non-holomorphic Saito-Kurokawa lift we construct a preimage under the vector-valued lowering operator. In analogy with the case of harmonic weak elliptic Maa{\ss} forms, this preimage allows for a natural decomposition into a…

We establish a correspondence between vector-valued modular forms with respect to a symmetric tensor representation and quasimodular forms. This is carried out by first obtaining an explicit isomorphism between the space of vector-valued…

We consider specific linear combinations of two loop modular graph functions on the toroidal worldsheet with $2s$ links for $s=2, 3$ and $4$. In each case, it satisfies an eigenvalue equation with source terms involving $E_{2s}$ and $E_s^2$…

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras.…

We develop a theory of modular forms on the groups $\mathrm{SO}(3,n+1)$, $n \geq 3$. This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and…

\textit{Weak moonshine} for a finite group $G$ is the phenomenon where an infinite dimensional graded $G$-module $$V_G=\bigoplus_{n\gg-\infty}V_G(n)$$ has the property that its trace functions, known as McKay-Thompson series, are modular…

Modular graph functions are $SL(2,{\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $\tau$. For one-loop graphs they reduce to real analytic Eisenstein series.…

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more…

The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such…

In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non--positive rational ends, with values in analytic functions…

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…

In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to…

The Kohnen plus space, roughly speaking, is a space consisting of modular forms of half integral weight with some property in Fourier coefficients. For example, the $n$-th coefficient of a normal modular form of weight $k+1/2$ in the plus…

We continue the study of the representation theory of a regular weak multiplier bialgebra with full comultiplication, started in arXiv:1306.1466, arXiv:1311.2730. Yetter-Drinfeld modules are defined as modules and comodules, with…

We formulate a notion of modular form on the double half-plane for half-integral weights and explain its relationship to the usual notion of modular form. The construction we provide is compatible with certain physical considerations due to…

This article is an overview of Zagier's and Kim's work on traces of singular moduli. We give more detailed or new proofs to some of their results and also describe some algorithms to compute spaces of Jacobi forms and weight $3/2$ modular…

Using techniques from the theory of mock modular forms and harmonic Maass forms, especially Weierstrass mock modular forms, we establish several dimension formulas for certain strongly rational, holomorphic vertex operator algebras,…