Fourier coefficients and slopes of Drinfeld modular forms

Abstract

Let [Formula: see text] be a Drinfeld modular form of level [Formula: see text] which is an eigenform for the Hecke operator [Formula: see text] ([Formula: see text] a prime of [Formula: see text]). We study the relations between the Fourier coefficients of [Formula: see text] and the [Formula: see text]-adic valuation of its eigenvalue (slope). We use formulas for some of the Fourier coefficients of [Formula: see text] to provide bounds and estimates on the slopes and, in particular, to find necessary conditions for “large” slopes, whose existence is closely connected with conjectures on oldforms and newforms.