Abhyankar's conjecture

Not to be confused with Abhyankar's lemma.

In abstract algebra, Abhyankar's conjecture for affine curves is a conjecture of Shreeram Abhyankar posed in 1957, on the Galois groups of algebraic function fields of characteristic p.^[1] The soluble case was solved by Serre in 1990^[2] and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater.^[3][4][5]

Statement

The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p.

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

π : C′ → C.

Geometrically, the assertion that π is ramified at a finite set S of points on C means that π restricted to the complement of S in C is an étale morphism. This is in analogy with the case of Riemann surfaces. 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.

Results

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).

Raynaud proved the case where C is the projective line over K, the conjecture stating that G can be realised as a Galois group of L, unramified outside S containing s + 1 points, if and only if

n ≤ s.

The general case was proved by Harbater. Where g is the genus of C, G can be realised if and only if

n ≤ s + 2 g.