Abhyankar's conjecture

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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

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.

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