On Congruences Between Drinfeld Modular Forms

Abstract

Let ${\mathbf{F}}_q$ denote a finite field of characteristic $p$ and let $n$ be an effective divisor on the affine line over ${\mathbf{F}}_q$ and let $v$ be a point on the affine line outside $n$. In this paper, we get congruences between ${\mathbb{Q}}_l$-valued weight two $v$-old Drinfeld modular forms and $v$-new Drinfeld modular forms of level $vn$. In order to do this, we shall first construct a cokernel torsion-free injection from a full lattice in the space of $v$-old Drinfeld modular forms of level $vn$ into a full lattice in the space of all Drinfeld modular forms of level $vn$. To get this injection we use ideas introduced by Gekeler and Reversat on uniformization of jacobians of Drinfeld moduli curves.

24 pages