Abhyankar's conjecture
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.[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 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
- ↑ 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" (in French), Comptes Rendus de l'Académie des Sciences, Série I 311 (6): 341–346
- ↑ Raynaud, Michel (1994), "Revêtements de la droite affine en caractéristique p > 0", Inventiones Mathematicae 116 (1): 425–462, doi:10.1007/BF01231568, Bibcode: 1994InMat.116..425R.
- ↑ Harbater, David (1994), "Abhyankar's conjecture on Galois groups over curves", Inventiones Mathematicae 117 (1): 1–25, doi:10.1007/BF01232232, Bibcode: 1994InMat.117....1H.
- ↑ 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
External links
- Weisstein, Eric W.. "Abhyankar's conjecture". http://mathworld.wolfram.com/AbhyankarsConjecture.html.
- A layman's perspective of Abhyankar's conjecture from Purdue University
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