Abhyankar's Conjecture

For a Finite Group $G$, let $p(G)$ be the Subgroup generated by all the Sylow p-Subgroup of $G$. If $X$ is a projective curve in characteristic $p>0$, and if $x_0$, ..., $x_t$ are points of $X$ (for $t>0$), then a Necessary and Sufficient condition that $G$ occur as the Galois Group of a finite covering $Y$ of $X$, branched only at the points $x_0$, ..., $x_t$, is that the Quotient Group $G/p(G)$ has $2g+t$ generators.

Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the projective line with a point deleted), and Harbater (1994) proved the full Abhyankar conjecture by building upon this special solution.

See also Finite Group, Galois Group, Quotient Group, Sylow p-Subgroup

References

Abhyankar, S. ``Coverings of Algebraic Curves.'' Amer. J. Math. 79, 825-856, 1957.

Harbater, D. ``Abhyankar's Conjecture on Galois Groups Over Curves.'' Invent. Math. 117, 1-25, 1994.

Raynaud, M. ``Revêtements de la droite affine en caractéristique $p>0$ et conjecture d'Abhyankar.'' Invent. Math. 116, 425-462, 1994.