On the Atkin $U_t$-operator for $\Gamma_1(t)$-invariant Drinfeld cusp forms

TL;DR

This paper investigates the diagonalizability of the Atkin $U_t$-operator on Drinfeld cusp forms for specific modular groups, revealing characteristic-dependent behavior and providing proofs for small weights.

Contribution

It demonstrates diagonalizability of $U_t$ in odd characteristic for small weights and analyzes non-diagonalizability in even characteristic using harmonic cocycles.

Findings

$U_t$ is diagonalizable in odd characteristic for weights $k extless= 2q$

Non-diagonalizability in even characteristic depends on antidiagonal blocks

Provides new insights into the structure of $U_t$-operator on Drinfeld cusp forms

Abstract

We study the diagonalizability of the Atkin $U_t$-operator acting on Drinfeld cusp forms for $\Gamma_1(t)$ and $\Gamma(t)$ using Teitelbaum's interpretation as harmonic cocycles. For small weights $k\leqslant 2q$, we prove $U_t$ is diagonalizable in odd characteristic and we point out that non diagonalizability in even characteristic depends on antidiagonal blocks.