The p-rank and ramification structure of covers of curves in characteristic p

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Abstract for NSF grant DMS-0701303 of PriesThe p-rank and ramification structure of covers of curves in characteristic p.Galois theory and number theory have 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. Number theory arose classically as a way of finding integer solutions to diophantine equations. There are modern applications of Galois theory and number theory to data-transfer codes. One of the goals of the PI is to increase activity in number theory in the Colorado region. The graduate students of the PI are integrally involved in the research in this proposal. The PI is a co-organizer of the new Front Range Number Theory Colloquium. This seminar has participants from at least five institutions in Colorado and Wyoming. It leads to increased research and communication about number theory in this geographic region.Galois covers and Jacobians of complex curves are well-understood subjects. In characteristic p, there are new phenomena that lead to major open problems on the topic of Galois covers and Jacobians of curves. These phenomena involve wildly ramified group actions on curves and the p-rank of curves. Let k be an algebraically closed field of characteristic p 0. Let C be a smooth connected projective k-curve of genus g. The p-rank of C is the integer f between 0 and g so that the number of p-torsion points on the Jacobian of C equals p raised to the power f. The PI proposes a research project about the p-rank and ramification structure of covers of k-curves. As applications, the PI plans to:1) determine the minimal genus of a G-Galois cover of the affine line for many groups G;2) determine the number of irreducible components of the moduli space of Artin-Schreier curves of genus g;3) show a generic curve with genus g and p-rank f has a-number 1 if fg;4) show there exists a smooth k-curve of genus g and p-rank f whose Jacobian is absolutely irreducible for every g 2 and every f between 0 and g.The arithmetic objects appearing in the proposal include ramification filtrations, norm groups, group schemes, and monodromy groups. The geometric techniques used for the proposal include deformation, formal patching, and stratifications of moduli spaces of curves.