Abhyankar's conjecture
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] 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]
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
- 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.
References
- ^ 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, doi:10.1007/BF01231568, Zbl 0798.14013.
- ^ Harbater, David (1994), "Abhyankar's conjecture on Galois groups over curves", Inventiones Mathematicae, 117 (1): 1–25, doi:10.1007/BF01232232, Zbl 0805.14014.
- ^ Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 11 (3rd ed.), Springer-Verlag, p. 70, ISBN 978-3-540-77269-9, Zbl 1145.12001