Abhyankar's conjecture

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In mathematics, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of function fields of characteristic p. This problem was solved in 1994 by work of Michel Raynaud and David Harbater.

The problem involves a finite group G, a prime number p, and an algebraic curve C, defined over a field K of characteristic p. Assume that C is non-singular, which can be done without changing the function field

K(C).

The question addresses the existence of Galois extensions L of K(C), with G as Galois group, and with restricted ramification. From a geometric point of view L corresponds to another curve C′, and a morphism

π : C′ → C.

Ramification geometrically, and by analogy with the case of Riemann surfaces, consists of a finite set S of points x on C, such that π restricted to the complement of S in C is an étale morphism. In Abhyankar's conjecture, S is fixed, and the question is what G can be. This is therefore a special type of inverse Galois problem.

The subgroup p(G) is defined to be the subgroup generated by all the Sylow subgroups of G for the prime number p. This is a normal subgroup, and the parameter n is defined as the minimum number of generators of

G/p(G).

Then for the case of C the projective line over K, the conjecture states that G can be realised as a Galois group of L, unramified outside S containing s + 1 points, if and only if

n ≤ s.

This was proved by Raynaud.

For the general case, proved by Harbater, let g be the genus of C. Then G can be realised if and only if

n ≤ s + 2 g.

[edit] References

Michel Raynaud, Revêtements de la droite affine en caractéristique p > 0, Invent. Math. 116 (1994) 425-462
David Harbater, Abhyankar's conjecture on Galois groups over curves, Invent. Math. 117 (1994) 1-25