Congruences between modular forms, Galois representations, and arithmetic consequences

现代数学的一个主要主题是研究物体如何变化或变形是有用的。例如，函数在某一点上的值只是一个数字，但是了解（连续）函数在该点的小邻域内的行为可以提供定性信息，例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，这样人们才能谈论“一点点”扰动一个物体。在数论中，人们经常用“p进”距离来代替通常的距离概念——一种测量一个数如何被给定素数p整除的距离。PI将研究“模形式”的p进变化——其图具有许多对称性的特殊函数。模形式变分的研究是数论领域的一个重要课题，它已经产生了许多深刻的成果，如费马大定理的证明。PI和她的合作者已经发现了p进变化的新方向，这些方向尚未被探索，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指导高中生，本科生和研究生，并组织区域和国际层面的会议。在更技术性的层面上，有三个主要的主题需要研究。前两个是关于模形式之间的同余，首先是在“令人烦恼”的情况下，然后是爱森斯坦同余。最后一个主题是关于伽罗瓦表现的图像。每个主题包含多个项目，既直接解决主题，也解决算术结果。上面提到的（tame） p进变的一个新方向出现在令人烦恼的设置中，PI和她的合作者可以在相关的Hecke代数上证明一个结构结果。对这一现象的深入研究将使她能够探索Iwasawa理论和Bloch-Kato猜想的温和类比的结果。在爱森斯坦同余的主题中，PI将在权值2和质数平方水平上探索伪变形环和Hecke代数的结构。这对于被整数的p次方根截断的字段的类群的p秩有很自然的应用。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乘积的图像，以一种表征欧拉系统机械所需元素可用性的方式来完成这项工作。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。

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.