相关论文: Exceptional theta functions and arithmeticity of m…

We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We…

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). In a previous paper, we gave an explicit form of the Fourier expansion…

Let $E$ be a cubic \'etale extension of the rational numbers which is totally real, i.e., $E \otimes \mathbf{R} \simeq \mathbf{R} \times \mathbf{R} \times \mathbf{R}$. There is an algebraic $\mathbf{Q}$-group $S_E$ defined in terms of $E$,…

We consider a special theta lift $\theta(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $\theta(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the…

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…

By using the degenerate Whittaker functions, we study the Fourier expansion of the Gan-Gurevich lifts which are Hecke eigen quaternionic cusp forms of weight $k$ ($k\geq 2$, even) on the split exceptional group $G_2$ over $\mathbb{Q}$ which…

For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic…

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the…

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…

The purpose of this partly expository paper is to give an introduction to modular forms on $G_2$. We do this by focusing on two aspects of $G_2$ modular forms. First, we discuss the Fourier expansion of modular forms, following work of…

Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic…

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, in…

In this article we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at…

In this paper we study the compatible family of degree-4 Scholl representations $\rho_{\ell}$ associated with a space $S$ of weight $\kappa> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It…

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…

Let $\mathbf{G}=\mathrm{U}(2,n)$ be the unitary group associated to a Hermitian space over a quadratic imaginary number field $E$. We assume that 2 is unramified in $E$, and the Hermitian space splits at all finite places and has signature…

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized…

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…

This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form \phi(\tau) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve…

We develop vanishing and cuspidality criteria for quaternionic modular forms on $G=\mathrm{Spin}(4,4)$ using a theory of scalar Fourier coefficients. By analyzing a Fourier-Jacobi expansion for these forms, we prove that a level one…