Small Mathieu group coverings in characteristic two

Abstract

Abstract. Explicit equations are given for unramified coverings of the affine line in characteristic two whose Galois groups are the Mathieu groups of degrees 11 and 12 and the automorphism group of the Mathieu group of degree 12.

Galois groups over function fields of positive characteristic

“…Abhyankar used a method that relied on a characterization of the Galois groups as permutation groups while Serre used a method based on Lüroth's theorem and the invariant theory of Dickson. Later, in [3], Abhyankar obtained the Mathieu group, M 23 , as the Galois group of x 23 + tx 3 + 1 / F 2 (t). His proof was based on an idea used by Serre (in Abhyankar and Yie, [5] ) who showed that P SL (3,2) is the Galois group of x 7 + tx 3 + 1 / F 2 (t).…”

Equations in characteristic 3 with Mathieu groups as their Galois groups

“…See Section 33 of [10] for more information about the universal covering group. In their paper ( [9]), Abhyankar and Yie proved irreducibility of certain polynomials by resolving singularities of plane curves. Similarly, in Section 2, we shall prove that F~ is irreducible in k(X) [Y] by resolving singularities of a projective plane curve.…”

“…In papers [5], [8] and [9], linearization lemmas were proved and these lemmas produced some upper bounds for the relevant Galois groups. These lemmas say that a polynomial f divides a linearized polynomial A.…”

“…By modifying the proofs of the linearization lemmas in [6], [8] and [9], we shall now prove the following lemma. …”

“…In support of this conjecture, he gave, in Propositions 2 and 3 and Theorem 3 of [2], three quite general families of equations giving unramified coverings of Lk and suggested to compute their Galois groups. Recently this task has been carried out for various subcases in [3] to [9]. In this paper we will consider equations which are slight variations of the family given in Proposition 2 of [2].…”