A Family of Étale Coverings of the Affine Line

Abstract

It this note we prove the following theorem. Letπalg1(A1C) be the algebraic fundamental group of the affine line overC, whereCis the completion of the algebraic closure ofFq((1/T)), andFqis a field withqelements. IfFqhas at least four elements, then we show that there is a continuous surjectionπalg1(A1C)→limSL2?(A/I)/{+ ±1}, whereA=Fq[T] and the inverse limit is over the family of non-zero, proper ideals ofA. This result is proved by using the moduli of Drinfel'dA-modules of rank two overCwithI-level structures; these curves give (tamely) ramified covers of the line and the tame ramification is removed using a variant of Abhyankar's lemma.