The Poisson kernel for Drinfeld modular curves

Abstract

In an earlier work [T 1], the author described a technique for constructing rigid analytic modular forms on the p-adic upper half plane by means of an integral transform (a Poisson Kernel). In this paper, we apply these methods to the study of rigid analytic modular forms on the upper half plane over a complete local field k of p. Arising out of the theory of Drinfeld modules [D], these modular forms, first studied by Goss (see [Gol, Go2]), are in many ways analogous to the classical modular forms for SL2 (Z). For a brief, but effective, introduction to the theory of these characteristic p modular forms, see the introduction to Gekeler's paper [Gekl]. Also useful is Gekeler's book [Gek2]. We establish three main results. First of all, we show that if, k is the function field of a complete, geometrically irreducible curve C over the finite field lFq with q elements, k is a completion of k, F is an arithmetic subgroup of GL2(k), and ,u is a measure on Pi coming from a harmonic cocycle of weight n for I, (see Definition 1 below) then