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

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.

This paper has been withdrawn and substituted with the two new ones arXiv:1710.01036 and arXiv:1710.01038 containing new developments