Abhyankar's conjecture

Given data
A finite group $$G$$, a prime number $$p$$, a nonsingular projective curve $$X$$, defined over a field $$K$$ of characteristic $$p$$. $$x_0, x_1, \ldots, x_t$$ (where $$t > 0$$) are points of $$X$$. Let $$g$$ denote the genus of $$X$$.

Statement
Let $$p(G)$$ denote the subgroup generated by all the $$p$$-Sylow subgroups of $$G$$. Then, the following are equivalent:


 * $$G$$ occurs as the Galois group of a branched covering $$Y$$ of $$X$$, branched only at the points $$x_0, \ldots, x_t$$
 * The quotient group $$G/p(G)$$ has $$2g+t$$ generators.

Progress towards the conjecture
Raynaud settled the conjecture in the affine case, and Harbater proved the full conjecture by building upon this special solution.