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

Abstract

This paper continues work of the earlier articles with the same title. For two classes of modular forms f: • para-Eisenstein series αk and • coefficient forms ak, where k ∈ N and a is a non-constant element of Fq[T], the growth behavior on the fundamental domain and the zero loci Ω(f) as well as their images BT (f) in the Bruhat-Tits building BT are studied. We obtain a complete description for f = αk and for those of the forms ak where k ≤ deg a. It turns out that in these cases, αk and ak are strongly related, e.g., BT (ak) = BT (αk), and that BT (αk) is the set of Qpoints of a full subcomplex of BT with nice properties. As a case study, we present in detail the outcome for the forms α2 in rank 3.