A Hecke-equivariant decomposition of spaces of Drinfeld cusp forms via representation theory, and an investigation of its subfactors

TL;DR

This paper extends the understanding of Drinfeld cusp forms by decomposing their spaces via representation theory, exploring subfactors, and providing numerical evidence of new obstructions to analogs of the Maeda conjecture.

Contribution

It generalizes Teitelbaum's isomorphism to an adelic setting, relates intertwining maps to hyperderivatives, and investigates dimension formulas for Hecke-invariant subquotients.

Findings

Discovered a new obstruction to the Maeda conjecture for Drinfeld forms.

Numerical data suggests an infinite supply of rational eigenforms with specific Hecke eigenvalues.

Extended Teitelbaum's isomorphism to arbitrary levels and adelic settings.

Abstract

There are various reasons why a naive analog of the Maeda conjecture has to fail for Drinfeld cusp forms. Focussing on double cusp forms and using the link found by Teitelbaum between Drinfeld cusp forms and certain harmonic cochains, we observed a while ago that all obvious counterexamples disappear for certain Hecke-invariant subquotients of spaces of Drinfeld cusp forms of fixed weight, which can be defined naturally via representation theory. The present work extends Teitelbaum's isomorphism to an adelic setting and to arbitrary levels, it makes precise the impact of representation theory, it relates certain intertwining maps to hyperderivatives of Bosser-Pellarin, and it begins an investigation into dimension formulas for the subquotients mentioned above. We end with some numerical data for $A=\mathbb{F}_3[t]$ that displays a new obstruction to an analog of a Maeda conjecture by discovering a conjecturally infinite supply of $\mathbb{F}_3(t)$-rational eigenforms with combinatorially given (conjectural) Hecke eigenvalues at the prime $t$.