In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p.[1] This problem was solved in 1994 by work of Michel Raynaud and David Harbater.[2][3]
The problem involves a finite group G, a prime number p, and the function field of nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p.
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
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
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
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