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 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).