Measuring Galois Actions and Moduli Spaces

算术几何领域侧重于研究由多项式方程定义的形状（称为变体）以及这些方程的解的数量。PI将研究曲线的性质和数论学家特别感兴趣的相关变化。该项目包括研究费马曲线（与著名的安德鲁·怀尔斯对费马大定理的证明有关），具有大量对称性的曲线（其中算术和几何之间的相互作用提供了丰富的结构），以及更一般曲线的算术不变量。PI还提出了两个项目来训练学生的数论。第一项是在2020年夏季对科罗拉多州立大学的本科生进行数学研究培训。第二个目标是举办一个视频会议，在全球范围内开放访问算术几何培训研讨会。这将培养该领域的研究生，并在该领域的研究人员之间建立联系。PI将继续指导和培训学生，并为数学界提供服务。PI计划研究3个课题：(1)费马曲线的伽罗瓦上同调：费马大定理的证明确定了费马曲线上定义在有理场q上的所有点。PI计划研究一个伽罗瓦上同调的映射，该映射测量有理点的阻塞，并证明该阻塞对于定义在环场上的费马曲线不消失。(2) jacobian和Pryms的特殊Shimura变体：对于g3属的曲线，Torelli态射的像在g维的abelian变体的模空间中是开密的。PI提出研究具有额外自同构的曲线族，其Torelli态射的像在相关的Shimura变体中是开密的；这些家庭被称为特殊家庭。PI建议发现和分析更多特殊的曲线族和Prym变种。PI还提出了一种非特殊家庭的结果推导方法，从特殊家庭的输入开始进行归纳。(3)伽罗瓦覆盖的p-扭转不变量的变化：在有限域上定义的椭圆曲线可以是普通曲线，也可以是超奇异曲线，如Hasse、Deuring和Igusa所研究的。对于g1，有一些不变量可以推广g属曲线或g维的阿贝尔变化曲线的超奇异性质。例如，牛顿多边形描述了上同调上的Frobenius算子的信息。测量这些不变量很重要，因为它们决定了曲线的算术和几何信息。确定这些不变量中的哪一个出现在曲线上是一个诱人的开放性问题。PI提出了许多关于牛顿多边形和曲线雅可比矩阵的p-扭转群格式的方案。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。

The field of arithmetic geometry focuses on the study of shapes (called varieties) that are defined by polynomial equations and the number of solutions to those equations. The PI will investigate properties of curves and related varieties that are of particular interest to number theorists. The project includes studying Fermat curves (connected to the celebrated proof of Fermat's Last Theorem by Andrew Wiles), curves with a large set of symmetries (for which the interplay between the arithmetic and the geometry provides a rich structure), and arithmetic invariants of more general curves. The PI also proposes two projects to train students in number theory. The first is to train undergraduates at Colorado State University in mathematics research during summer 2020. The second is to host a video-conference training seminar in arithmetic geometry, with world-wide open access. This will train graduate students in this field and build connections among researchers in this area. The PI will continue to mentor and train students and to provide service to the math community. The PI plans to study 3 topics: (1) Galois cohomology of Fermat curves: The proof of Fermat's Last Theorem determined all of the points on the Fermat curves that are defined over the rational field Q. The PI proposes to study a map in Galois cohomology that measures an obstruction for rational points and to show that this obstruction does not vanish for the Fermat curves defined over cyclotomic fields. (2) Special Shimura varieties for Jacobians and Pryms: For curves of genus g 3, the image of the Torelli morphism is open and dense in the moduli space of abelian varieties of dimension g. The PI proposes to study families of curves with extra automorphisms for which the image of the Torelli morphism is open and dense in an associated Shimura variety; these families are called special. The PI proposes to find and analyze more special families of curves and Prym varieties. The PI also proposes a method to deduce results about non-special families, working inductively starting with the input of special families. (3) Variation of p-torsion invariants for Galois covers: An elliptic curve defined over a finite field can be ordinary or supersingular, as studied by Hasse, Deuring, and Igusa. For g 1, there are invariants that generalize the supersingular property for a curve of genus g or an abelian variety of dimension g. For example, the Newton polygon characterizes information about the Frobenius operator on the cohomology. Measuring these invariants is important because they determine arithmetic and geometric information about the curve. It is a tantalizing open question to determine which of these invariants occur for curves. The PI proposes numerous projects about the Newton polygon and p-torsion group scheme of Jacobians of curves.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.