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

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

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

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 is larger or equal to , and the forms in question include the coefficient forms and the Eisenstein series ().
Paper Structure (5 sections, 11 theorems, 91 equations)

This paper contains 5 sections, 11 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Modular forms and their expansions
  3. The special Eisenstein series $E_{q^{k}-1}$
  4. The coefficient forms $g_{k}$
  5. The $h$-function

Key Result

Theorem 1.8

Theorems & Definitions (20)

  • Remark 1.6
  • Theorem 1.8: Gekeler17, BassonBreuerPink24
  • Theorem 1.11: Gekeler88, Gekeler25
  • Theorem 1.12: Gekeler85, Basson17, Gekeler25
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • proof
  • Theorem 2.8
  • ...and 10 more