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Dimension formulas for certain spaces of Drinfeld cusp forms

Gebhard Boeckle, Peter Mathias Graef, Iason Papadopoulos

TL;DR

This work determines dimension formulas for spaces of Drinfeld cusp forms linked to $ ext{SL}_2(A)$-invariant harmonic cocycles valued in absolutely irreducible representations, using Brauer-character techniques to replace prior methods. By constructing a filtration and leveraging the Teitelbaum residue isomorphism, it reduces the problem to computing $d_{k,q}= ext{dim}_F C_{ m har}(L_k)^{ ext{SL}_2(A)}$ for $A= ext{F}_q[t]$ and derives explicit formulas in small $q$ cases, as well as a general asymptotic formula as $\dim_F L_k$ grows. The paper provides detailed Brauer-character computations for equal-characteristic representations, including the Steinberg module, and expresses multiplicities via inner products over $p$-regular conjugacy classes; it demonstrates that simple closed forms exist only in limited cases (e.g., $q=2,3,5$) and become increasingly intricate for larger $q$. The asymptotic result shows linear growth of the dimension with a universal constant depending on $q$ and the parity of $p$, offering important insight into the distribution of Drinfeld cusp forms in large weight. These results extend prior work in BGP21 and contribute a Brauer-character framework for understanding Drinfeld modular forms in equal characteristic.

Abstract

In this short note, we derive dimension formulas for spaces of Drinfeld cusp forms corresponding to harmonic cocycles invariant under the group $\mathrm{SL}_2(\mathbb{F}_q[t])$ and with values in absolutely irreducible $\mathrm{SL}_2(\mathbb{F}_q(t))$-representations via the theory of Brauer characters. This generalizes results in [BGP21] obtained by different methods. In addition, we prove a simple asymptotic formula for these dimensions.

Dimension formulas for certain spaces of Drinfeld cusp forms

TL;DR

This work determines dimension formulas for spaces of Drinfeld cusp forms linked to -invariant harmonic cocycles valued in absolutely irreducible representations, using Brauer-character techniques to replace prior methods. By constructing a filtration and leveraging the Teitelbaum residue isomorphism, it reduces the problem to computing for and derives explicit formulas in small cases, as well as a general asymptotic formula as grows. The paper provides detailed Brauer-character computations for equal-characteristic representations, including the Steinberg module, and expresses multiplicities via inner products over -regular conjugacy classes; it demonstrates that simple closed forms exist only in limited cases (e.g., ) and become increasingly intricate for larger . The asymptotic result shows linear growth of the dimension with a universal constant depending on and the parity of , offering important insight into the distribution of Drinfeld cusp forms in large weight. These results extend prior work in BGP21 and contribute a Brauer-character framework for understanding Drinfeld modular forms in equal characteristic.

Abstract

In this short note, we derive dimension formulas for spaces of Drinfeld cusp forms corresponding to harmonic cocycles invariant under the group and with values in absolutely irreducible -representations via the theory of Brauer characters. This generalizes results in [BGP21] obtained by different methods. In addition, we prove a simple asymptotic formula for these dimensions.

Paper Structure

This paper contains 9 sections, 22 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Recollections on splitting fields and Brauer characters
  3. Conjugacy classes of the group SLTwoFq
  4. Equal characteristic representations of the group SLTwoFq
  5. Brauer characters of certain SLTwoFq-Reps-representations
  6. Computations for small q
  7. The cases q=2,3,5
  8. The cases q=4,7,11
  9. Asymptotic behavior

Key Result

Theorem 1

Let $A={\mathbb{F}}\,\!{}_q[t]$ and write $d_{k,q}:=\dim_F C_\mathrm{har}(L_{k})^{{\rm SL}_2(A)}$. Then

Theorems & Definitions (42)

  • Theorem : cf. \ref{['p5']}, \ref{['p3']} and \ref{['p2']}
  • Definition 1
  • Proposition 1: CR81
  • Proposition 2: CR81
  • Proposition 3: CR81
  • Theorem 2.1: Webb
  • Proposition 4
  • Remark 1
  • Definition 2
  • Definition 3
  • ...and 32 more