Misplaced Pages

Abhyankar's conjecture

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Abhyankar conjecture) Not to be confused with Abhyankar's lemma.

In abstract algebra, Abhyankar's conjecture is a conjecture of Shreeram Abhyankar posed in 1957, 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.

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 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.

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

n ≤ s + 2 g.

References

  1. Abhyankar, Shreeram (1957), "Coverings of Algebraic Curves", American Journal of Mathematics, 79 (4): 825–856, doi:10.2307/2372438, JSTOR 2372438.
  2. Serre, Jean-Pierre (1990), "Construction de revêtements étales de la droite affine en caractéristique p", Comptes Rendus de l'Académie des Sciences, Série I (in French), 311 (6): 341–346, Zbl 0726.14021
  3. Raynaud, Michel (1994), "Revêtements de la droite affine en caractéristique p > 0", Inventiones Mathematicae, 116 (1): 425–462, Bibcode:1994InMat.116..425R, doi:10.1007/BF01231568, Zbl 0798.14013.
  4. Harbater, David (1994), "Abhyankar's conjecture on Galois groups over curves", Inventiones Mathematicae, 117 (1): 1–25, Bibcode:1994InMat.117....1H, doi:10.1007/BF01232232, Zbl 0805.14014.
  5. 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

External links

Categories: