Modular Forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\): \(t\)-expansions of the basic forms

TL;DR

This paper derives explicit formulas for initial coefficients of fundamental modular forms associated with \\mathrm{GL}(r, \\mathbb{F}_{q}[T]) for ranks r ≥ 3, including coefficient forms and Eisenstein series.

Contribution

It provides the first explicit formulas for the initial expansion coefficients of key modular forms in higher rank function field settings.

Findings

Closed formulas for initial coefficients of basic forms

Explicit expressions for coefficient forms and Eisenstein series

Enhances understanding of modular forms over function fields

Abstract

We give closed formulas for the first few expansion coefficients of the basic modular forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\). Here the rank \(r\) is larger or equal to \(3\), and the forms in question include the coefficient forms \(g_{1}, \dots, g_{r}\) and the Eisenstein series \(E_{q^{i}-1}\) (\(i \in \mathbb{N}\)).