Moduli spaces for wildly ramified covers of curves

Colorado State University, Fort Collins CO

Investigators

Abstract

Project Summary for award DMS-0400461 of Pries The theory of Galois covers of the complex projective line (Riemann sphere) with given branch locus is well-understood; namely, one can describe the inertia groups of such a cover with Riemann's Existence Theorem and study families of such covers via their moduli spaces, which are called Hurwitz spaces. These techniques extend to covers of the projective line over an algebraically closed field k as long as the ramification is tame. However, if the ramification is wild, i.e. if the characteristic of the field k divides the order of some inertia group, new phenomena occur which are much less understood. There are major open problems involving the inertia groups and deformation theory of wildly ramified covers. The PI proposes to study wildly ramified covers of curves, with the goal of answering some of these problems; in particular, the PI expects to compute the dimension of the deformation space of a cover in terms of its ramification invariants. The PI also proposes to construct moduli spaces for wildly ramified Galois covers of curves in characteristic p and to investigate some of their properties. Galois theory has high appeal to a broad audience. Several open problems in this area can be explained to non-mathematicians and the topic connects diverse areas of math. Galois theory arose classically as a means of understanding symmetries of equations and of classifying extensions of the rational numbers. Some of the early applications were that it is impossible to trisect an angle, double the volume of a cube, or solve a quintic equation. The PI proposes to develop a new course on Galois theory, to introduce graduate students to the fundamental background and active research problems of this topic. The PI would like to continue to lead research programs for students; the program which the PI led in 2002 motivated several students (including students from underrepresented groups) to pursue graduate study in mathematics. An application of Galois theory in characteristic $p$ to data-transfer codes was recently discovered. These codes can be constructed using curves of low degree which have many points defined over finite fields. The PI would like to develop the connection between Galois theory and coding theory further. Finally, the PI will continue collaborating with mathematicians in Germany and New York and will disseminate the results from this research through conferences and papers.