Evaluating Actions, Obstructions, and Reductions for Covers of Curves

该项目将重点研究具有罕见属性的曲线，通过建立基础和产生新的应用。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领域之间的联系。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中开放猜想的虚拟研讨会。为了使研讨会更容易获得和持久，PI将组织VaNTAGe活动来培训研究生并开发VaNTAGe YouTube频道。PI将在数学方面指导学生，例如，作为科罗拉多州立大学高中生索尼娅·科瓦列夫斯基日的指导老师。更准确地说，PI将评估伽罗瓦作用、上同调障碍和曲线的超奇异缩减。PI的第一个研究项目将得到伽罗瓦群对具有自同构曲线上同调的作用的结果。这个主题的重要性在于，它将揭示有理数领域的绝对伽罗瓦群，正如Grothendieck， Anderson和Ihara的工作所理解的那样。为了完成这个项目，PI将评估Belyi曲线的上同性图，包括分类图、海侵图和障碍物图。在第二个研究项目中，PI将推广Elkies的结果，证明某些四属曲线具有无限多个超奇异还原素数。该项目将集中于在四个点上分支的投影线的特殊循环覆盖族，以及它们相关的Hurwitz空间和酉Shimura变种。该策略将使用均匀化、二次型、复杂乘法和约简技术。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的超奇异曲线的数量。该项目将阐明Oort关于任意属超奇异曲线存在的猜想，并将Eichler-Deuring质量公式推广到循环盖的Hurwitz空间的情况。这些项目的一个统一主题是，它们将展示具有自同构曲线的几何和算术技术之间丰富的相互作用。该奖项反映了美国国家科学基金会的法定使命，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。

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.