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THE MINIMAL MODULAR FORM ON QUATERNIONIC
$E_{8}$
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 2 / March 2022
- Published online by Cambridge University Press:
- 20 August 2020, pp. 603-636
- Print publication:
- March 2022
-
- Article
- Export citation
-
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 
$\unicode[STIX]{x1D703}_{Gan}$ on quaternionic 
$E_{8}$ and some applications. The 
$Sym^{8}(V_{2})$-valued automorphic function 
$\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, 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 
$\operatorname{GSp}_{2n}$.
