Generators and Relations for the Étale Fundamental Group

If C is a smooth curve over an algebraically closed field k of characteristic exponent p, then the structure of the maximal prime-to-p quotient of the étale fundamental group is known by analytic methods. In this paper, we discuss the properties of the fundamental group that can be deduced by purely algebraic techniques. We describe a general reduction from an arbitrary curve to the projective line minus three points, and show what can be proven unconditionally about the maximal pro-nilpotent and pro-solvable quotients of the prime-to-p fundamental group. Included is an appendix which treats the tame fundamental group from a stack-theoretic perspective.