Can a Drinfeld Module be Modular?

Let k be a global function field with field of constants F_r, r = p^m, and let ∞ be a fixed place of k. In his habilitation thesis [4], Gebhard B¨ockle attaches abelian Galois representations to characteristic p valued cusp eigenforms and double cusp eigenforms [20] such that Hecke eigenvalues correspond to the image of Frobenius elements. In the case where k = F_r(T ) and ∞ corresponds to the pole of T, it then becomes reasonable to ask whether rank 1 Drinfeld modules over k are themselves “modular” in that their Galois representations arise from a cusp or double cusp form. This paper gives an introduction to [4] with an emphasis on modularity and closes with some specific questions raised by B¨ockle’s work.