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

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 $\mathcal{B}\mathcal{T}(f)$ in the Bruhat-Tits building $\mathcal{B}\mathcal{T}$ are studied. We obtain a complete description for $f={\alpha }_k$ and for those of the forms ${}_a{\ell }_k$ where $k\le °a$. It turns out that in these cases, ${\alpha }_k$ and ${}_a{\ell }_k$ are strongly related, e.g., $\mathcal{B}\mathcal{T}({}_a{\ell }_k)=\mathcal{B}\mathcal{T}({\alpha }_k)$, and that $\mathcal{B}\mathcal{T}({\alpha }_k)$ is the set of $\mathbb{Q}$-points of a full subcomplex of $\mathcal{B}\mathcal{T}$ 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}

34 pages