Moduli of curves in positive characteristic: stratifications and filtrations

PI提出研究与特征为p的代数闭域k上定义的曲线相关的分层和过滤。p-秩和a-数是k曲线雅可比矩阵的p-扭转群格式的不变量；这些不变量引起曲线模空间的分层。建议的第一部分是分析这些地层的几何形状，包括不可约性、尺寸和边界的问题。第二部分是关于k上仿射线的伽罗瓦覆盖和野惯性群的分支过滤。这些过滤在回答关于曲线的大分枝覆盖的抬升和变形的问题中起着关键作用。这些项目都需要几何技术（退化到模空间的边界，变形/提升/形式修补）。然而，该建议的所有部分都是关于k曲线的p群覆盖的算术，并涉及算术对象（群方案，Frobenius的作用，Galois群，范数群，形式群）。此外，给出了具有给定自同构群或p-扭转不变量的特征p有限域上曲线存在性的具体应用。伽罗瓦理论是作为理解方程对称性和对复数子域进行分类的一种经典方法而产生的。数论是作为寻找多项式方程整数解的一种经典方法而出现的。PI的研究是关于有限域中由多项式方程和系数定义的曲线上的函数。本主题对密码系统和数据传输代码有一些应用。PI将为研究生主持几个关于曲线和伽罗瓦封面的夏季研究研讨会。目的是给学生合作研究的经验，并增加与本提案相关的问题的知识。PI还参与了其他具有更广泛影响的倡议，包括：共同组织WIN（妇女人数）倡议，以增加妇女在数论方面的研究培训；亚利桑那冬季学校的联合组织；以及科罗拉多州立大学数学系和哥斯达黎加大学之间的新联系。

The PI proposes to study stratifications and filtrations associated with curves defined over an algebraically closed field k of characteristic p. The p-rank and a-number are invariants of the p-torsion group scheme of the Jacobian of a k-curve; these invariants induce stratifications of moduli spaces of curves. The first part of the proposal is to analyze the geometry of such strata, including questions about irreducibility, dimension, and boundaries. The second part of the proposal is about Galois covers of the affine line over k and the ramification filtrations of the wild inertia groups. These filtrations play a key role in answering questions about lifting and deformation of wildly ramified covers of curves. These projects all require geometric techniques (degeneration to boundaries of moduli spaces, deformation/lifing/formal patching). Yet all parts of the proposal are about the arithmetic of p-group covers of k-curves and involve arithmetic objects (group schemes, action of Frobenius, Galois groups, norm groups, formal groups). In addition, the projects yield concrete applications about the existence of curves over a finite field of characteristic p with given automorphism group or p-torsion invariants.Galois theory arose classically as a means of understanding symmetries of equations and of classifying subfields of the complex numbers. Number theory arose classically as a way of finding integer solutions to polynomial equations. The PI's research is about functions on curves defined by polynomial equations with coefficients in a finite field. This topics has some applications to cryptosystems and data-transfer codes. The PI will lead several summer research workshops for graduate students about curves and Galois covers. The goal is to give the students experience with collaborative research and to increase knowledge about problems relevant to this proposal. The PI is also involved with other initiatives that have broader impacts including: co-organization of WIN (women in numbers) initiatives to increase the research training of women in number theory; co-organization of the Arizona Winter School; and a new liaison between the math departments of CSU and the Universidad de Costa Rica.