Abhyankar's conjecture

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Not to be confused with Abhyankar's lemma.

In

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

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. doi:10.2307/2372438
    .
  2. Zbl 0726.14021
  3. Zbl 0798.14013
    .
  4. Zbl 0805.14014
    .
  5. Zbl 1145.12001

External links
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