Abhyankar's Conjecture

For a finite group G, let p(G) be the subgroup generated by all the Sylow p-subgroups 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

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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.", 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.

Cite this as:

Weisstein, Eric W. "Abhyankar's Conjecture." From MathWorld--A Wolfram Web Resource.

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