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THE MINIMAL MODULAR FORM ON QUATERNIONIC $E_{8}$

Part of: Discontinuous groups and automorphic forms Linear algebraic groups and related topics

Published online by Cambridge University Press:  20 August 2020

Aaron Pollack Open the ORCID record for Aaron Pollack [Opens in a new window]
Aaron Pollack*
Affiliation:
Department of Mathematics, Duke University, Durham, NC, USA ([email protected])
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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 $\unicode[STIX]{x1D703}_{Gan}$ 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 $\operatorname{GSp}_{2n}$.

Keywords

minimal representationmodular formsexceptional groupsarithmetic invariant theoryEisenstein series

MSC classification

Primary: 11F03: Modular and automorphic functions
Secondary: 11F30: Fourier coefficients of automorphic forms 20G41: Exceptional groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 2 , March 2022 , pp. 603 - 636
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

The author has been supported by the Simons Foundation via Collaboration Grant number 585147.

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