In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater.
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
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.
- Abhyankar, Shreeram (1957), "Coverings of Algebraic Curves", American Journal of Mathematics, 79 (4): 825–856, doi:10.2307/2372438.
- Serre, Jean-Pierre (1990), "Construction de revêtements étales de la droite affine en caractéristique p", C. R. Acad. Sci., Paris, Sér. I (in French), 311 (6): 341–346, Zbl 0726.14021
- Raynaud, Michel (1994), "Revêtements de la droite affine en caractéristique p > 0", Inventiones Mathematicae, 116 (1): 425–462, Zbl 0798.14013, doi:10.1007/BF01231568.
- Harbater, David (1994), "Abhyankar's conjecture on Galois groups over curves", Inventiones Mathematicae, 117 (1): 1–25, Zbl 0805.14014, doi:10.1007/BF01232232.
- Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 11 (3rd ed.), Springer-Verlag, p. 70, ISBN 978-3-540-77269-9, Zbl 1145.12001