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

Ernst-Ulrich Gekeler

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

Number Theory 2026-01-06 v1

Authors: Ernst-Ulrich Gekeler

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 $E_{q^{i}-1}$ ( $i \in \mathbb{N}$ ).

Keywords

modular forms eisenstein series special functions and series expansions

Cite

@article{arxiv.2601.02034,
  title  = {Modular Forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\): \(t\)-expansions of the basic forms},
  author = {Ernst-Ulrich Gekeler},
  journal= {arXiv preprint arXiv:2601.02034},
  year   = {2026}
}

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