Abhyankar's conjecture

Statement

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.