Étale Covers of Affine Spaces in Positive Characteristic

More Étale Covers of Affine Spaces in Positive Characteristic

We prove that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, étale away from the hyperplane H at infinity, which maps a chosen divisor into H and some chosen smooth points not on the divisor to points not in H . This improves an earlier result of the author, which was restricted to infinite perf...

A family of étale coverings of the affine line

This note was inspired by a colloquium talk given by S. S. Abhyankar at the Tata Institute, on the work of Abhyankar, Popp and Seiler (see [2]). It was pointed out in this talk that classical modular curves can be used to construct (by specialization) coverings of the affine line in positive characteristic. In this “modular” optic it seemed natural to consider Drinfel’d modular curves for const...

Insufficiency of the Brauer-manin Obstruction Applied to Étale Covers

Let k be any global field of characteristic not 2. We construct a k-variety X such that X(k) is empty, but for which the emptiness cannot be explained by the BrauerManin obstruction or even by the Brauer-Manin obstruction applied to finite étale covers.

Covers of the affine line in positive characteristic with prescribed ramification

Let k be an algebraically closed field of characteristic p > 0. In this note we consider Galois covers g : Y → P1k which are only branched at t = ∞. We call such covers unramified covers of the affine line. It follows from Abhyankar’s conjecture for the affine line proved by Raynaud ([7]) that such a cover exists for a given group G if and only if G is a quasi-p group, i.e. can be generated by ...

Convergence of an Implicit Iteration for Affine Mappings in Normed and Banach Spaces