Evaluating Actions, Obstructions, and Reductions for Covers of Curves

该项目将通过建立基础和产生新的应用，专注于具有稀有性质的曲线的研究。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，Vantage研讨会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活动培训研究生，并开发Vantage YouTube频道。例如，PI将作为科罗拉多州立大学高中生索尼娅·科瓦列夫斯基日的教师顾问，指导学生的数学。更准确地说，PI将计算伽罗瓦作用量、上同调障碍和曲线的超奇异约化。PI的第一个研究项目将产生关于伽罗瓦群在具有自同构的曲线的上同调上同调上的作用的结果。这个话题的重要性在于，它将阐明有理数领域的绝对伽罗瓦群，正如Grothendieck、Anderson和Ihara所理解的那样。为了完成这个项目，PI将评估Belyi曲线的上同调地图，包括分类地图、海侵地图和障碍地图。在第二个研究项目中，PI将通过证明亏格4的某些曲线有无穷多个超奇异约化素数来推广Elkies的一个结果。这个项目将集中在射影直线在四点上分支的特殊的循环覆盖族，以及它们相关的Hurwitz空间和酉Shimura簇。该策略将使用统一化、二次型、复数乘法和归约技术。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。这个项目将阐明Oort关于存在任意亏格的超奇异曲线的猜想，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，它们将展示具有自同构的曲线的几何和算术技术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智力优势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.