Congruences between modular forms, Galois representations, and arithmetic consequences

模形式、伽罗瓦表示和算术结果之间的同余

• 批准号: 2301738
• 负责人: Jaclyn Lang
• 金额: $ 17.5万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301738&HistoricalAwards=false
• 关键词: Congruences; between; modular; forms; Galois

A major theme in modern mathematics is that it is useful to study how objects can be changed or deformed. For instance, the value of a function at a point is just one number, but knowledge of how a (continuous) function behaves in a tiny neighborhood of that point gives qualitative information like whether the function is increasing or decreasing and how fast. This requires a notion of distance so that one can talk about perturbing an object “a little bit”. In number theory, one often replaces the usual notion of distance with a “p-adic” distance — one that measures how divisible a number is by a given prime number p. The PI will investigate the p-adic variation of “modular forms” — special functions whose graphs have many symmetries. Studying variation of modular forms is a topic in number theory that has led to many deep results such as the proof of Fermat’s Last Theorem. The PI and her collaborators have found new directions of p-adic variation that have yet to be explored and promise to yield new insights about arithmetic. This investigation will incorporate young scientists at many career stages as the PI mentors high school, undergraduate, and graduate students and organizes conferences at both a regional and international level. On a more technical level, there are three overarching topics to be studied. The first two concern congruences between modular forms, first in the “vexing” setting and then Eisenstein congruences. The final topic concerns images of Galois representations. Each topic contains multiple projects, both addressing the topic directly as well as arithmetic consequences. One new direction of (tame) p-adic variation referenced above occurs in the vexing setting, where the PI and her collaborators can prove a structure result on the relevant Hecke algebra. A deeper study of this phenomenon will allow her to explore consequences for tame analogues of Iwasawa theory and the Bloch-Kato Conjectures. Within the topic of Eisenstein congruences, the PI will explore the structure of pseudodeformation rings and Hecke algebras in weight 2 and prime-square level. This has natural applications to the p-rank of the class group of the field cut out by a p-th root of an integer. Finally, she will use her recent work with Conti and Medvedovsky on images of 2-dimensional Galois representations to understand images of finite products of such representations, doing this in a way that characterizes the availability of elements needed for Euler system machinery.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.

现代数学的一个主要主题是研究物体如何改变或变形是有用的。 例如，一个函数在一个点上的值只是一个数字，但是关于一个（连续）函数在该点的一个微小邻域中的行为的知识可以给出定性的信息，比如函数是增加还是减少，以及速度有多快。 这需要一个距离的概念，这样我们就可以谈论“一点点”扰动一个物体。 在数论中，人们经常用“p-adic”距离来代替通常的距离概念-测量一个数被给定素数p整除的程度。PI将研究“模形式”的p-adic变化-特殊函数的图形具有许多对称性。 研究模形式的变化是数论中的一个课题，它导致了许多深刻的结果，例如费马大定理的证明。 PI和她的合作者已经发现了p-adic变化的新方向，这些方向还有待探索，并有望产生关于算术的新见解。 这项调查将包括许多职业阶段的年轻科学家作为PI导师高中，本科和研究生，并在区域和国际层面组织会议。在技术层面上，有三个主要课题需要研究。 前两个关注的一致性之间的模块形式，首先在“烦恼”的设置，然后爱森斯坦一致性。 最后一个主题涉及伽罗瓦表示的图像。每个主题包含多个项目，既直接解决主题，也解决算术后果。 上面提到的（驯服）p-adic变化的一个新方向出现在令人烦恼的设置中，其中PI和她的合作者可以证明相关Hecke代数上的结构结果。 对这一现象的深入研究将使她能够探索岩泽理论和布洛赫-加藤猜想的驯服类似物的后果。 在爱森斯坦同余的主题内，PI将探索伪变形环和Hecke代数在权重2和素数平方水平上的结构。 这有自然的应用程序的p-秩的类组的领域削减了p次根的一个整数。 最后，她将利用她最近与Conti和Medvedovsky在二维伽罗瓦表示的图像上的工作来理解这种表示的有限产品的图像，这样做的方式表征欧拉系统机械所需的元素的可用性。这个奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。