Moduli of curves in positive characteristic: stratifications and filtrations

PI建议研究与定义在特征为p的代数闭域k上的曲线有关的分层和滤子。p-秩和a-数是k-曲线的Jacobian的p-扭群格式的不变量；这些不变量导致曲线的模空间的分层。该提案的第一部分是分析这些地层的几何结构，包括关于不可约、尺寸和边界的问题。第二部分是关于k上仿射线的Galois覆盖和野惯性群的分支滤子。这些滤子在回答有关曲线的广泛分叉覆盖的提升和变形的问题时起着关键作用。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，该方案的所有部分都是关于k-曲线的p-群覆盖的算法，并且涉及算术对象(群方案、Frobenius的作用、Galois群、范数群、形式群)。此外，这些方案还给出了特征为p的有限域上具有给定的自同构群或p-挠不变量的曲线的存在性的具体应用。伽罗瓦理论是作为一种理解方程对称性和对复数的子域进行分类的经典方法而产生的。数论是作为寻找多项式方程的整数解的一种经典方法出现的。PI的研究是关于由有限域上系数的多项式方程所定义的曲线上的函数。本主题在密码系统和数据传输码方面有一些应用。PI将为研究生举办几个关于曲线和伽罗瓦封面的暑期研究研讨会。目标是让学生体验合作研究，并增加对与该提案相关的问题的了解。国际和平研究所还参与了具有更广泛影响的其他倡议，包括：共同组织WIN(妇女在数字中)倡议，以增加妇女在数论方面的研究培训；共同组织亚利桑那州冬季学校；以及在CSU和哥斯达黎加大学的数学系之间建立新的联络人。

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.