2002/10/31 by David Goss
Mathematics · #math.NT #math.AG #msc:11F52
published as Journal of the Ramanujan Math. Soc. {\bf 17} No. 4 (2002) 221-260 · Final corrected version
arxiv created 2003/01/05 · arxiv updated 2009/11/30
Let k be a global function field with field of constants \Fr and let ∞ be a fixed place of k. In his habilitation thesis \citeboc2, Gebhard Böckle attaches abelian Galois representations to characteristic p valued cusp eigenforms and double cusp eigenforms \citego1 such that Hecke eigenvalues correspond to the image of Frobenius elements. In the case where k=\Fr(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 \citeboc2 with an emphasis on modularity and closes with some specific questions raised by Böckle's work.