On the Atkin $U_t$-operator for ${\Gamma }_1(t)$-invariant 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 $k⩽2q$, we prove $U_t$ is diagonalizable in odd characteristic and we point out that non diagonalizability in even characteristic depends on antidiagonal blocks.

It contains parts of the previous arXiv:1702.08801 [math:NT], which has now been split in two papers with some new results