Evaluating Actions, Obstructions, and Reductions for Covers of Curves

评估曲线覆盖的动作、障碍和缩减

• 批准号: 2200418
• 负责人: Rachel Pries
• 金额: $ 27.03万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200418&HistoricalAwards=false
• 关键词: Evaluating; Actions; Obstructions; Reductions; Covers

This project will focus on the study of curves with rare properties, by both building foundations and yielding new applications. The PI will study geometric, arithmetic, and algebraic structures of these curves, thereby developing connections between the areas of arithmetic geometry, algebraic number theory, and Galois theory. The PI will also build connections in the mathematical community by developing the VaNTAGe seminar, a virtual seminar about open conjectures in number theory and arithmetic geometry. To make the seminar more accessible and long-lasting, the PI will organize VaNTAGe activities to train graduate students and develop the VaNTAGe YouTube channel. The PI will mentor students in mathematics, for example, as the faculty advisor for the Sonia Kovalevsky day for high school students at Colorado State University. More precisely, the PI will evaluate Galois actions, cohomological obstructions, and supersingular reductions of curves. The first research project of the PI will yield results about the action of Galois groups on the cohomology of curves with automorphisms. The importance of this topic is that it will shed light on the absolute Galois group of the field of rational numbers, as understood by the work of Grothendieck, Anderson, and Ihara. To do this project, the PI will evaluate maps in cohomology for Belyi curves, including the classifying map, the transgression map, and obstruction maps. In the second research project, the PI will generalize a result of Elkies by proving that certain curves of genus four have infinitely many primes of supersingular reduction. This project will be centered on special families of cyclic covers of the projective line branched at four points, and their associated Hurwitz spaces and unitary Shimura varieties. The strategy will use uniformization, quadratic forms, complex multiplication, and reduction techniques. In the third research project, the PI will count the number of supersingular curves in one-dimensional families of curves with automorphisms. This project will shed light on conjectures of Oort about the existence of supersingular curves of arbitrary genus and will generalize the Eichler-Deuring mass formula to the case of Hurwitz spaces of cyclic covers. A unifying theme of the projects is that they will demonstrate the rich interplay between geometric and arithmetic techniques for curves with automorphisms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将通过建立基础和产生新的应用，重点研究具有罕见特性的曲线。PI将研究这些曲线的几何，算术和代数结构，从而发展算术几何，代数数论和伽罗瓦理论之间的联系。PI还将通过开发Vantage研讨会在数学界建立联系，这是一个关于数论和算术几何开放式课程的虚拟研讨会。 为了使研讨会更容易和持久，PI将组织Vantage活动，以培训研究生，并开发Vantage YouTube频道。PI将指导学生的数学，例如，作为教师顾问的索尼娅科瓦列夫斯基一天的高中生在科罗拉多州立大学。 更准确地说，PI将评估伽罗瓦行动，上同调障碍和超奇异约化曲线。PI的第一个研究项目将产生关于伽罗瓦群对具有自同构的曲线的上同调的作用的结果。这个主题的重要性在于它将阐明有理数领域的绝对伽罗瓦群，正如格罗滕迪克、安德森和伊原的工作所理解的那样。 为了完成这个项目，PI将评估Belyi曲线的上同调图，包括分类图，海侵图和障碍图。 在第二个研究项目中，PI将通过证明亏格为4的某些曲线具有无限多个超奇异约化素数来推广Elkies的结果。 这个项目将集中在特殊的家庭循环覆盖的投影线分支在四个点，和他们相关的赫维茨空间和酉志村品种。 该策略将使用均匀化，二次形式，复数乘法和减少技术。 在第三个研究项目中，PI将计算具有自同构的一维曲线族中超奇异曲线的数量。 这个项目将阐明奥尔特关于超奇异曲线的存在性的假设，并将Eichler-Deuring质量公式推广到循环覆盖的Hurwitz空间的情况。 这些项目的一个统一主题是，它们将展示几何和算术技术与自同构曲线之间的丰富相互作用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。