On Drinfeld modular forms of higher rank V: The behavior of distinguished forms on the fundamental domain

TL;DR

This paper investigates the growth, zero loci, and images in the Bruhat-Tits building of certain Drinfeld modular forms of higher rank, providing detailed descriptions and relations among these forms on the fundamental domain.

Contribution

It offers a complete analysis of para-Eisenstein series and coefficient forms for specific parameters, revealing their relationships and geometric properties in the Bruhat-Tits building.

Findings

Complete description of growth and zero loci for selected forms.

Identification of relationships between different classes of forms.

Detailed case study of rank 3 forms, especially .

Abstract

\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 $k \leq \deg a$. 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}