Let ∇ be a meromorphic connection on a vector bundle over a compact Riemann surface Γ. An automorphism σ : Γ → Γ is called a symmetry of ∇ if the pull-back bundle and the pull-back connection can be identified with ∇. We study the symmetries by means of what we call the Fano Group; and, under the hypothesis that ∇ has a unimodular reductive Galois group, we relate the differential Galois group,...

this paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical galois theory and involves generalized central extensions, commutators, and internal groupoids in barr exact mal’tsev and more general categories. galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...

As noted in [Jon97], only examples of solvable Galois actions on classes of dessins d'enfants have been explicitly constructed. In this paper an action of the nonsolvable alternating group A5, represented as a Galois group, on a class of elliptic dessins is presented.

Unramified coverings of the affine line in characteristic two are constructed having alternating groups as Galois groups. The proof uses Jacobson’s criterion for the Galois group of an equation to be contained in the alternating group. Alternative proofs use the Berlekamp discriminant or the Revoy discriminant. These are related to the Arf invariant.

Let X be a smooth projective connected curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Let G be a finite group, P a Sylow p-subgroup of G and NG(P ) its normalizer in G. We show that if there exists an étale Galois cover Y → X with group NG(P ), then G is the Galois group wan étale Galois cover Y → X , where the genus of X depends on the order of G, th...

Galois groups of infinite p-extensions of number fields unramified at p are a complete mystery. We find by computer a family of pro-p groups that satisfy everything that such a Galois group must, and give evidence for the conjecture that these are the only such groups. This suggests that these mysterious Galois groups indeed have a specific form of presentation. There are surprising connections...

This is the final paper in a series of four, concerning the surface T × T embedded in CP 8 , where T is a the one dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto CP 2 , and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover...

This article studies the Albanese kernel TF (E × E), for an elliptic curve E over a p-adic field F . The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from TF (E ×E)/p to the Galois cohomology group H(F, E[p] ⊗ E[p]), for E/F ordinary, without requiring that the ptorsion points are F -rational, or even that the Galois module E[p] is ...

The paper develops an Artin-Schreier theory for valued elds: mutatis mutandis, valuations on any eld F are encoded in the absolute Galois group G F of F , just like orderings. We present a Galois characterization of henselian elds as well as a purely group theoretic characterization of decomposition subgroups of G F for arbitrary elds F. The approach is elementary and self-contained.

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1, t2, . . . , tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.