Curves, covers, and cohomology

算术几何的一个基本主题是多项式方程的解的存在性，其系数位于复数以外的域。从历史上看，具有超对称性的方程一直是研究的焦点，因为相应几何对象的自同构提供了额外的结构，有助于阐明解的集合。PI的研究是关于有限域中系数多项式方程定义的曲线上的函数。我们的目标是分析连接到这些曲线的代数结构的不变量，以确定这些不变量如何在参数空间的曲线变化，并调查发生异常结构。作为该提案的更广泛影响，PI将与柯林斯堡发现博物馆合作，为儿童建立有关密码学的活动套件。PI还参与了具有更广泛影响的其他举措：PI是WIN网络的领导者，其目标是通过新的研究合作来振兴女性在数论方面的研究事业; PI是亚利桑那州冬季学校的共同组织者2010-2015。 曲线和阿贝尔品种表现出新的现象，积极的特征p.这些现象通常出现由于潜在的存在态射的程度p.他们导致重要的上同调不变量，如p-秩，牛顿多边形，和埃克达尔-奥尔特型。PI的建议的目标是证明：（非）-存在的结果约阿贝尔品种和曲线与给定的不变量;和结构的结果有关的分层引起的不变量对各种模空间。特别是，PI将研究超奇异阿贝尔簇和雅可比矩阵;自同构和p-挠不变量之间的相互作用;公式，类似于Deuring-Shafarevich公式，用于在分歧数据方面更精细的Ekedahl-Oort类型不变量的变化;以及非代数闭域上曲线的有理点。关键技术包括：交叉因子的模空间;退化到奇异曲线的紧凑型;结果约monodromy的家庭曲线;上同调计算（德拉姆或决议）;和行动的Frobenius和Vershiebung。

An underlying theme of arithmetic geometry is the existence of solutions to polynomial equations whose coefficients lie in fields other than the complex numbers. Historically, equations having extra symmetry have been a focus of inquiry since the automorphisms of the corresponding geometric object provide extra structure which helps illuminate the set of solutions. The PI's research is about functions on curves defined by polynomial equations with coefficients in a finite field. The goal is to analyze invariants of algebraic structures attached to these curves, to determine how these invariants vary across parameter spaces for the curves, and to investigate which exceptional structures occur. As a broader impact of the proposal, the PI will collaborate with the Fort Collins Museum of Discovery to build activity kits about cryptography for children. The PI is also involved with other initiatives that have broader impacts: the PI is a leader of the WIN network, whose goal is to vitalize the research careers of women in number theory via new research collaborations; and the PI was a co-organizer of the Arizona Winter School from 2010-2015. Curves and abelian varieties exhibit new phenomena in positive characteristic p. These phenomena typically arise due to the underlying presence of morphisms of degree p. They lead to important cohomological invariants, such as the p-rank, Newton polygon, and Ekedahl-Oort type. The goal of the PI's proposal is to prove: (non)-existence results about abelian varieties and curves with given invariants; and structural results about the stratifications induced by the invariants on various moduli spaces. In particular, the PI will investigate supersingular abelian varieties and Jacobians; the interplay between automorphisms and p-torsion invariants; formulae, akin to the Deuring-Shafarevich formula, for the variation in more refined invariants of the Ekedahl-Oort type in terms of ramification data; and rational points of curves over non-algebraically closed fields. Key techniques include: intersection of divisors on moduli spaces; degeneration to singular curves of compact type; results about monodromy of families of curves; cohomology calculations (de Rham or of resolutions); and actions of Frobenius and Vershiebung.