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On Drinfeld modular forms of higher rank V: The behavior of distinguished forms on the fundamental domain
Journal of Number Theory ( IF 0.7 ) Pub Date : 2020-12-01 , DOI: 10.1016/j.jnt.2020.10.007
Ernst-Ulrich Gekeler

\begin{document} \begin This paper continues work of the earlier articles with the same title. For two classes of modular forms $f$: \begin{itemize} \item para-Eisenstein series $\alpha_{k}$ and \item coefficient forms ${}_a \ell_{k}$, where $k \in \mathbb{N}$ and $a$ is a non-constant element of $\mathbb{F}_{q}[T]$, \end{itemize} the growth behavior on the fundamental domain and the zero loci $\Omega(f)$ as well as their images $\mathcal{BT}(f)$ in the Bruhat-Tits building $\mathcal{BT}$ are studied. We obtain a complete description for $f = \alpha_{k}$ and for those of the forms ${}_{a}\ell_{k}$ where $k \leq °a$. It turns out that in these cases, $\alpha_{k}$ and ${}_{a}\ell_{k}$ are strongly related, e.g., $\mathcal{BT}({}_{a}\ell_{k}) = \mathcal{BT}(\alpha_{k})$, and that $\mathcal{BT}(\alpha_{k})$ is the set of $\mathbb{Q}$-points of a full subcomplex of $\mathcal{BT}$ with nice properties. As a case study, we present in detail the outcome for the forms $\alpha_{2}$ in rank 3. \end{abstract} \maketitle \end{document}

中文翻译:

关于更高阶 V 的 Drinfeld 模形式:可区分形式在基本域上的行为

\begin{document} \begin 本文延续了之前同名文章的工作。对于两类模形式 $f$:\begin{itemize} \item para-Eisenstein 系列 $\alpha_{k}$ 和 \item 系数形式 ${}_a \ell_{k}$,其中 $k \in \ mathbb{N}$ 和 $a$ 是 $\mathbb{F}_{q}[T]$ 的非常量元素,\end{itemize} 基本域和零位点上的增长行为 $\Omega (f)$ 以及他们在 Bruhat-Tits 大楼 $\mathcal{BT}$ 中的图像 $\mathcal{BT}(f)$ 进行了研究。我们获得了 $f = \alpha_{k}$ 和 ${}_{a}\ell_{k}$ 形式的完整描述,其中 $k \leq °a$。事实证明,在这些情况下,$\alpha_{k}$ 和 ${}_{a}\ell_{k}$ 是强相关的,例如 $\mathcal{BT}({}_{a}\ell_ {k}) = \mathcal{BT}(\alpha_{k})$, 而 $\mathcal{BT}(\alpha_{k})$ 是 $\mathcal{BT}$ 的具有良好性质的完整子复杂的 $\mathbb{Q}$-points 的集合。作为案例研究,我们详细介绍了排名 3 中表格 $\alpha_{2}$ 的结果。 \end{abstract} \maketitle \end{document}
更新日期:2020-12-01
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