On the Atkin $U_t$-operator for $\Gamma_0(t)$-invariant Drinfeld cusp forms

TL;DR

This paper investigates the diagonalizability of the Atkin $U_t$-operator on Drinfeld cusp forms for $ ext{Gamma}_0(t)$, providing explicit eigenvalue computations, characterizing diagonalizability in different characteristics, and proposing conjectures supported by numerical evidence.

Contribution

It offers new insights into the diagonalizability of the Atkin $U_t$-operator on Drinfeld cusp forms, including explicit eigenvalue calculations and conjectures on slopes and diagonalizability.

Findings

$U_t$ is diagonalizable in odd characteristic for small weights.

$U_t$ is not diagonalizable in even characteristic for odd weights.

Numerical evidence supports conjectures on slopes and diagonalizability.

Abstract

We study the diagonalizability of the Atkin $U_t$-operator acting on Drinfeld cusp forms for $\Gamma_0(t)$: starting with the slopes of eigenvalues and then moving to the space of cusp forms for $\Gamma_1(t)$ to use Teitelbaum's interpretation as harmonic cocycles which makes computations more explicit. We prove $U_t$ is diagonalizable in odd characteristic for (relatively) small weights and explicitly compute the eigenvalues. In even characteristic we show that it is not diagonalizable when the weight is odd (except for the trivial cases) and prove some cases of non diagonalizability in even weight as well. We also formulate a few conjectures, supported by numerical search, about diagonalizability of $U_t$ and the slopes of its eigenforms.