On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms

TL;DR

This paper investigates when the Atkin U-operator for Drinfeld cusp forms is diagonalizable, providing proofs for small weights, explicit eigenvalues, and conjectures about non-diagonalizability in certain cases.

Contribution

It offers new results on the diagonalizability of the U-operator, including explicit eigenvalues and a conjecture supported by numerical evidence and special case proofs.

Findings

U is diagonalizable for small weights

Explicit eigenvalues are computed

Conjecture on non-diagonalizability in even characteristic

Abstract

We study the diagonalizability of the Atkin $U$-operator acting on Drinfeld cusp forms for $\Gamma_1(t)$ and $\Gamma(t)$ using Teitelbaum's interpretation as harmonic cocycles. We prove $U$ is diagonalizable for small weights and explicitly compute the eigenvalues. We also formulate a conjecture, supported by numerical search and proofs in some special cases, about non diagonalizability of $U$ in even characteristic.