Abhyankar's conjecture
From Groupprops
This article is about a conjecture in the following area in/related to group theory: algebraic geometry. View all conjectures and open problems
This article or section of article is sourced from:Mathworld
Contents
Statement
Given data
A finite group , a prime number
, a nonsingular projective curve
, defined over a field
of characteristic
.
(where
) are points of
. Let
denote the genus of
.
Statement
Let denote the subgroup generated by all the
-Sylow subgroups of
. Then, the following are equivalent:
-
occurs as the Galois group of a branched covering
of
, branched only at the points
- The quotient group
has
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.