Curves, covers, and cohomology

曲线、覆盖和上同调

• 批准号: 1502227
• 负责人: Rachel Pries
• 金额: $ 15.45万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-15 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502227&HistoricalAwards=false
• 关键词: Curves; covers; cohomology

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.

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中。从历史上看，具有额外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有助于照亮解集合的额外结构。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。我们的目标是分析这些曲线上的代数结构的不变量，确定这些不变量如何在曲线的参数空间中变化，并调查哪些特殊结构会发生。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，为儿童制作关于密码学的活动工具包。国际和平协会还参与了其他具有更广泛影响的倡议：国际和平协会是WIN网络的领导者，其目标是通过新的研究合作振兴女性在数论方面的研究生涯；国际和平协会在2010-2015年间是亚利桑那州冬季学校的联合组织者。曲线和交换簇在正特征p中表现出新的现象。这些现象通常是由于p次态射的潜在存在而产生的。它们导致了重要的上同调不变量，如p-秩、牛顿多边形和Ekedahl-Oort型。PI建议的目的是证明：(非)-关于给定不变量的交换簇和曲线的存在性结果；以及关于各种模空间上由不变量引起的分层的结构性结果。特别是，PI将研究超奇异阿贝尔变种和雅可比；自同构和p-扭转不变量之间的相互作用；公式，类似于Deuring-Shafarevich公式，根据分支数据，用于更精细的Ekedahl-Oort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：模空间上因子的交集；紧致型奇异曲线的退化；曲线族单行的结果；上同调计算(de Rham或归结)；Frobenius和Vershiebung的作用。