The minimal modular form on quaternionic $E_8$

TL;DR

This paper explicitly computes the Fourier expansion of the minimal modular form on quaternionic E_8 and explores its applications to constructing special modular forms on related exceptional groups.

Contribution

It provides the Fourier expansion of the minimal modular form on quaternionic E_8 and introduces applications to other quaternionic exceptional groups.

Findings

Fourier expansion of the minimal modular form on quaternionic E_8 derived.

Construction of special modular forms on quaternionic E_7, E_6, and G_2.

Discussion of degenerate Heisenberg Eisenstein series as analogues to Siegel Eisenstein series.

Abstract

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 of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\theta_{Gan}$ on quaternionic $E_8$, and some applications. The $Sym^{8}(V_2)$-valued automorphic function $\theta_{Gan}$ is a weight four, level one modular form on $E_8$, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_7, E_6$ and $G_2$. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\mathrm{GSp}_{2n}$.