On the Vanishing and Cuspidality of $D_4$ Modular Forms

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.

💡 Research Summary

The paper studies quaternionic modular forms on the split spin group (G=\mathrm{Spin}(4,4)) (type (D_4)) and establishes precise criteria for both vanishing and cuspidality in terms of scalar Fourier coefficients. The author begins by recalling the definition of quaternionic modular forms as vector‑valued automorphic functions annihilated by the Schmid differential operator associated to a specific real representation of (G(\mathbb R)). For a level‑one form of weight (\ell>0) the scalar Fourier coefficients (\Lambda_\varphi