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

摘要：美国国家科学基金会资助DMS-0701303的Priesp-秩和分支结构的覆盖曲线的特征p，伽罗瓦理论和数论有很高的吸引力，以广泛的观众。 几个开放的问题，在这方面可以解释非数学家和主题连接不同领域的数学。伽罗瓦理论出现经典的一种手段，了解对称性的方程和分类扩展的有理数。 数论作为一种寻找丢番图方程整数解的方法而出现。 伽罗瓦理论和数论在数据传输代码中有现代应用。 PI的目标之一是增加科罗拉多地区数论的活动。 PI的研究生整体参与了本提案的研究。 PI是新的前线数论讨论会的共同组织者。 这个研讨会有来自科罗拉多和怀俄明州至少五个机构的参与者。 这导致了该地理区域数论研究和交流的增加。复杂曲线的伽罗瓦覆盖和雅可比矩阵是很好理解的主题。 在特征p，有新的现象，导致主要的公开问题的主题伽罗瓦覆盖和雅可比曲线。 这些现象涉及曲线上的广泛分歧的群作用和曲线的p秩。 设k是特征为p 0的代数闭域. 设C是亏格为g的光滑连通射影k-曲线。 C的p-秩是0和g之间的整数f，使得C的雅可比矩阵上的p-扭点的数量等于p的f次幂。 PI提出了一个关于k-曲线覆盖的p-秩和分支结构的研究项目。 作为应用，PI计划：1）确定多个群G的仿射线的G-Galois覆盖的最小亏格;2）确定亏格为g的Artin-Schreier曲线的模空间的不可约分支的个数;3）证明亏格为g且p秩f的一般曲线的个数为1，如果fg; 4）证明了存在亏格为g和p的光滑k曲线。秩f，其雅可比矩阵对于每个g 2和0到g之间的每个f都是绝对不可约的。提案中出现的算术对象包括分歧滤子，范数群，群方案，和monodromy组。 几何技术用于建议包括变形，正式修补，和分层的模空间的曲线。

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.