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Measuring Galois Actions and Moduli Spaces

测量伽罗瓦作用和模空间

基本信息

批准号：
1901819
负责人：
Rachel Pries
金额：
$ 17.63万
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：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901819&HistoricalAwards=false
关键词：
Measuring Galois Actions Moduli Spaces

项目摘要

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.

算术几何领域专注于研究由多项式方程定义的形状(称为簇)以及这些方程的解的数目。PI将研究数学家特别感兴趣的曲线和相关变种的性质。该项目包括研究费马曲线(与安德鲁·怀尔斯著名的费马最后定理的证明有关)、具有大量对称性的曲线(算术和几何之间的相互作用为其提供了丰富的结构)，以及更一般曲线的算术不变量。国际学生联合会还提出了两个项目来训练学生的数论。第一个是在2020年夏天对科罗拉多州立大学的本科生进行数学研究方面的培训。二是举办一次全球开放的算术几何视频会议培训研讨会。这将培养这一领域的研究生，并在这一领域的研究人员之间建立联系。国际数学社将继续指导和培训学生，并为数学界提供服务。PI计划研究三个主题：(1)Fermat曲线的Galois上同调：Fermat最后定理的证明确定了定义在有理域Q上的Fermat曲线上的所有点。PI建议研究Galois上同调中的一个映射，该映射度量有理点的障碍，并证明对于定义在割圆域上的Fermat曲线，这种障碍不会消失。(2)关于Jacobian和Pryms的特殊的Shimura簇：对于亏格g3的曲线，Torelli态射的像在g维交换簇的模空间中是开的和稠密的.PI建议研究具有额外自同构的曲线族，对于这些曲线族，Torelli态射的像在相伴随的Shimura簇中是开的和稠密的，称为特殊的.PI建议寻找和分析更多特殊的曲线族和Prym变种。PI还提出了一种推导非特殊家庭结果的方法，即从特殊家庭的输入开始归纳工作。(3)Galois覆盖的p-挠不变量的变化：定义在有限域上的椭圆曲线可以是普通的，也可以是超奇异的，正如Hasse，Deuring和Igusa所研究的那样。对于g1，存在推广了亏格为g的曲线或维为g的阿贝尔变换的超奇异性质的不变量。例如，牛顿多边形刻画了关于上同调上的Frobenius算子的信息。测量这些不变量很重要，因为它们确定有关曲线的算术和几何信息。这是一个诱人的悬而未决的问题，以确定这些不变量中的哪些出现在曲线上。PI提出了许多关于曲线的雅可比人的牛顿多边形和p-扭群方案的项目。这个奖项反映了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。

项目成果

期刊论文数量（12）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Every $BT_1$ group scheme appears in a Jacobian

每个 $BT_1$ 组方案都出现在雅可比行列式中

DOI：
10.1090/proc/15681
发表时间：
2022
期刊：
Proceedings of the American Mathematical Society
影响因子：
1
作者：
Pries, Rachel;Ulmer, Douglas
通讯作者：
Ulmer, Douglas

Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

全局函数域上广义勒让德曲线雅可比行列式的显式算术

DOI：
10.1090/memo/1295
发表时间：
2020
期刊：
Memoirs of the American Mathematical Society
影响因子：
0
作者：
Berger, Lisa;Hall, Chris;Pannekoek, René;Park, Jennifer;Pries, Rachel;Sharif, Shahed;Silverberg, Alice;Ulmer, Douglas
通讯作者：
Ulmer, Douglas

the de Rham cohomology of the Suzuki curves

铃木曲线的德拉姆上同调