On Drinfeld modular forms of higher rank II. (Q2055933)

This paper is a continuation of [the author, J. Théor. Nombres Bordx. 29, No. 3, 875--902 (2017; Zbl 1411.11041)] and is part of a series of at least five papers of the author on Drinfeld modular forms of higher rank. In the previous work, several results that were not essentially needed for that paper were stated without proof, but since now these facts are crucial, complete proofs are given here. In Theorems 2.4 and 2.6 it is shown that if \(f\) is an invertible holomorphic function on the Drinfeld symmetric space \(\Omega^r\) on which \(\Gamma:=\mathrm{GL}(r,{\mathbb F}_q[T])\), \(r\geq 3\), acts, then \(\log_q |f(\omega)|\) is constant on fibers of the building map \(\lambda\colon\Omega^r\to{\mathcal{BT}}\) to the Bruhat-Tits building and, regarded as a function on the set \({\mathcal{BT}}({\mathbb Q})\) of points of \({\mathcal{BT}}\) with rational barycenter coordinates, this function is affine. These results are used to study the vanishing loci of modular forms and to determine their images in \( {\mathcal{BT}}\) (Theorems 4.2, 4.5 and 4.8). In this last theorem, it is shown that the \(k\)-th para-Eiseinstein series \(\alpha_k\) is a locally uniform limit of coefficient forms \({_a\ell}_k\), when the degree of \(a\in {\mathbb F}_q[T]\) tends to infinity.