Abstract
We define oldforms and newforms for Drinfeld cusp forms of level t and conjecture that their direct sum is the whole space of cusp forms. Moreover we describe explicitly the matrix U associated to the action of the Atkin operator on cusp forms of level t and use it to compute tables of slopes of eigenforms. Building on such data, we formulate conjectures on bounds for slopes, on the diagonalizability of
and on various other issues. Via the explicit form of the matrix U we are then able to verify our conjectures in various cases (mainly in small weights).
Akwnoledgements
We would like to thank the anonymous referee for his/her prompt report and for informing us of the ongoing work of G. Böckle, P. Graef and R. Perkins on Maeda’s conjecture.
Notes
1 The anonymous referee kindly informed us that there is some ongoing work by G. Böeckle, P. Graef and R. Perkins on suitable formulations of Maeda’s conjecture in the Drinfeld setting.
2 There is quite a difference between our notations and the one in [Citation28, Ch IV, Lemma 4], but we could not find a more suitable reference and, in our opinion, our computations are clearer with our notations.