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arXiv 1810.04595math.NTmath.RT
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四元数$E_8$上的极小模形式

The minimal modular form on quaternionic $E_8$

Aaron Pollack

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摘要：

设$G$是$\mathbf{Q}$上的一个单约化群，具有例外Dynkin型和$G(\mathbf{R})$四元数（在Gross-Wallach意义下）。在前一篇文章中，我们给出了$G$上模形式沿Heisenberg抛物的幂幺根的Fourier展开的一个显式形式。本文给出了四元数$E_8$上的极小模形式$θ_{Gan}$的Fourier展开式及其应用. $Sym^{8}(V_2)$值自守函数$θ_{Gan}$是$E_8$上的一个权4级1模形式，GAN已经研究过.我们给出的应用是四元数$E_7, E_6$和$G_2$上特殊模形式的构造。我们还讨论了$G$群上的一类退化Heisenberg Eisenstein级数，它可以被认为是$\mathrm{GSp}_{2n}$群上的全纯Siegel Eisenstein级数的四元例外群的一个类似物.

英文摘要：

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 $θ_{Gan}$ on quaternionic $E_8$, and some applications. The $Sym^{8}(V_2)$-valued automorphic function $θ_{Gan}$ 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 $\mathrm{GSp}_{2n}$.

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