2017/10/03 by Andrea Bandini, Bandini, Andrea, Maria Valentino +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1710.01038
It contains parts of the previous arXiv:1702.08801 [math:NT], which has now been split in two papers with some new results
arxiv created 2017/10/03 · arxiv updated 2017/10/04
We study the diagonalizability of the Atkin Ut-operator acting on Drinfeld cusp forms for Γ0(t): starting with the slopes of eigenvalues and then moving to the space of cusp forms for Γ1(t) to use Teitelbaum's interpretation as harmonic cocycles which makes computations more explicit. We prove Ut 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 Ut and the slopes of its eigenforms.