P-adic Variation of Modular Galois Representations

该奖项涉及代数数论，这是对具有有理系数的多项式方程的解的研究，以及伽罗瓦作用，这是这些解之间的对称性。现代数论的一个主要主题是利用伽罗瓦动作来获得关于多项式方程的整数或有理解问题的新见解，这些问题自古以来就激发了数学活动。伽罗瓦行为用于解决这些问题的一个主要方法是将它们插入到不断变化的家庭中。打个比方，通过伽罗瓦动作进行插值可以被认为是将一根绳子穿过一组珠子。这个项目关注这些家族中的“简并性”或“奇点”，类似于绳子上的一个结。这个项目的目的不仅是“解开”这种简并性，以获取它们似乎模糊的信息，而且还揭示简并性本身中额外的数论信息。该项目的资金还将用于支持宾夕法尼亚州西部的数学活动，例如为匹兹堡数论日带来外部扬声器，并鼓励学生参与研究和推广活动。就学生研究而言，PI将为研究生和本科生研究人员提供建议，以实现该项目的目标研究成果。在拓展方面，PI将招募并支持本科生参与资助的拓展工作，以丰富中小学生的数学教育。伽罗瓦表示的p进变型和模形式的发展推动了现代代数数论的巨大进步。但是，当插值出现退化时，概念和工具就缺乏或需要改进。这个项目的目的是解决和应用这些退化在不同的设置使用同构工具。目标成果包括伽罗瓦表示和模形式之间的联系的改进，p-根插值模形式的新概念在朗兰兹对应的派生充实猜想中的应用，以及计算多项式方程的有理或积分解的新技术。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。

This award concerns Algebraic number theory, which is the study of solutions to polynomial equations with rational coefficients, and Galois actions, which are symmetries among these solutions. A major theme of modern number theory is to use Galois actions to gain new insight into questions about integer or rational solutions to polynomial equations that have stimulated mathematical activity since ancient times. One major way that Galois actions are applied toward these questions is to interpolate them into continuously varying families. To make an analogy, interpolation through the Galois actions can be thought of as threading a string through a set of beads. This project concerns "degeneracies" or "singularities" within these families, analogous to a knot lying at a point of convergence among strands of the string. This project aims to not only "untie" such degeneracies to access the information they seem to obscure, but also to reveal the additional number-theoretic information in the degeneracy itself. Funding for this project will also be dedicated to supporting mathematical activity in Western Pennsylvania, such as bringing external speakers to Pittsburgh Number Theory Days and encouraging student activity in research and outreach. As far as student research, the PI will advise graduate and undergraduate student researchers working toward the targeted research outcomes of this project. And as far as outreach, the PI will recruit and support undergraduate students working in grant-funded outreach efforts to enrich math education for elementary and middle school students. Developments in the p-adic variation of Galois representations and of modular forms has fueled great progress in modern algebraic number theory. But when degeneracies occur in interpolation, notions and tools are lacking or need refinement. This project aims to resolve and apply these degeneracies in various settings using homological tools. Among the targeted outcomes are refinements of links between Galois representations and modular forms, applications of new notions of p-adically interpolated modular forms to conjectures about derived enrichments of the Langlands correspondence, and new techniques to compute rational or integral solutions to polynomial equations.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.