On the Structure and Slopes of Drinfeld Cusp Forms

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 ${\mathbf{U}}_t$ 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 ${\mathbf{U}}_t$ 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).

Keywords:

• Drinfeld cusp forms
• Atkin operator
• newforms and oldforms
• slopes of eigenforms
• Fricke involution

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 [28, Ch IV, Lemma 4], but we could not find a more suitable reference and, in our opinion, our computations are clearer with our notations.

Additional information

Funding

M. Valentino has been supported by an “Ing. G. Schirillo” fellowship of INdAM.