For a finite group, let be the subgroup generated
by all the Sylow p-subgroups of . If is a projective curve in characteristic , and if , ..., are points of (for ), then a necessary and
sufficient condition that occur as the Galois group
of a finite covering of , branched only at the points , ..., , is that the quotient group has 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.
Abhyankar, S. "Coverings of Algebraic Curves." Amer. J. Math.79, 825-856, 1957.American Mathematical
Society. "Notices of the AMS, April 1995, 1995 Frank Nelson Cole Prize in Algebra."
http://www.ams.org/notices/199504/prize-cole.pdf.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 et conjecture d'Abhyankar." Invent. Math.116,
425-462, 1994.
, let 
be the subgroup generated
by all the Sylow p-subgroups of 
. If 
is a projective curve in characteristic 
, and if 
, ..., 
are points of 
(for 
), then a necessary and
sufficient condition that 
occur as the Galois group
of a finite covering 
of 
, branched only at the points 
, ..., 
, is that the quotient group 
has 
generators.
et conjecture d'Abhyankar." Invent. Math. 116,
425-462, 1994.