On the Vanishing and Cuspidality of $D_4$ Modular Forms

TL;DR

This paper establishes criteria for vanishing and cuspidality of quaternionic modular forms on certain groups, using Fourier coefficient analysis, and characterizes specific subspaces and growth conditions for these forms.

Contribution

It introduces new vanishing and cuspidality criteria for quaternionic modular forms via Fourier coefficients and characterizes Pollack's subspace with linear relations among primitive coefficients.

Findings

Vanishing of forms characterized by primitive Fourier coefficients.

Pollack's quaternionic Saito-Kurokawa subspace characterized by linear relations.

Cuspidality linked to polynomial growth of Fourier coefficients in weights ≥ 5.

Abstract

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 quaternionic modular form on $G$ vanishes if and only if its primitive Fourier coefficients are zero. Using this criterion, we characterize Pollack's quaternionic Saito-Kurokawa subspace by imposing a system of linear relation among certain primitive Fourier coefficients. This characterization strengthens earlier work of the author with Johnson-Leung, Negrini, Pollack, and Roy. We also study quaternionic modular forms in the more general setting of a group $G_J$ associated to a cubic norm structure $J$. Here we establish a new relationship between the degenerate Fourier coefficients of quaternionic modular forms, and the Fourier coefficients of the holomorphic modular forms associated to their constant terms. As a consequence, we prove that in weights $\ell\geq 5$, a level one quaternionic modular form on $G$ is cuspidal if and only if its non-degenerate Fourier coefficients satisfy a polynomial growth condition.