Related papers: The minimal modular form on quaternionic $E_8$

We discover a non-trivial relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers. This relation yields a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$, where $N > 1$ is…

We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL(n), Sp(4), and G_2, we perform an analysis of…

In a recent preprint, we constructed a sesquiharmonic Maass form $\mathcal{G}$ of weight $\frac{1}{2}$ and level $4N$ with $N$ odd and squarefree. Extending seminal work by Duke, Imamo\={g}lu, and T\'{o}th, $\mathcal{G}$ maps to Zagier's…

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished…

Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of…

In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simply-laced group. Let $\pi$ be a minimal or next-to-minimal automorphic representation of $G$. We prove that any $\eta\in \pi$ is…

Starting with a primitive Dirichlet character of conductor $N$, we construct a paramodular Siegel Eisenstein series of level $N^2$ and weight $k\geq4$. We calculate the Fourier expansion of the holomorphic Siegel modular form thus…

Let $V$ be a simple, rational, $C_2$-cofinite vertex operator algebra and $G$ a finite group acting faithfully on $V$ as automorphisms, which is simply called a rational vertex operator algebra with a $G$-action. It is shown that the…

We introduce a new family of real analytic modular forms on the upper half plane. They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated…

Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac-Moody groups. In particular, we analyse the Eisenstein series on $E_9(R)$, $E_{10}(R)$…

This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest 'Drinfeld modular…

In a recent work, we found formulas for the Fourier coefficients of automorphic forms of type $G_2$: holomorphic Siegel modular forms on $\mathrm{Sp}_6$ that are theta lifts from $G_2^c$, and cuspidal quaternionic modular forms on split…

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we…

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 undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit…

We prove that the space of cuspidal quaternionic modular forms on the groups of type $F_4$ and $E_n$ have a purely algebraic characterization. This characterization involves Fourier coefficients and Fourier-Jacobi expansions of the cuspidal…

We show that the quaternionic discrete series on G=Sp(1,1) with minimal K-type of dimension n+1 can be realized inside the space of Fueter-regular functions on the quaternionic ball B in H, with values in H^n. We then consider the…

We consider the minimal representation of the adjoint split group $E_7$ over a p-adic field. The representation has a model in a space of functions on a 17 dimensional cone $\Omega$, and elements of the unique parabolic subgroup Q with…

We generalize the modular invariance approach to include the half-integral weight modular forms. Accordingly the modular group should be extended to its metaplectic covering group for consistency. We introduce the well-defined half-integral…

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define…