P-adic Variation of Modular Galois Representations

该奖项涉及代数数论，这是研究的解决方案多项式方程的合理系数，和伽罗瓦行动，这是这些解决方案之间的对称性。现代数论的一个主要主题是使用伽罗瓦作用来获得关于多项式方程的整数或有理解的新见解，这些多项式方程自古以来就激发了数学活动。一个主要的方式，伽罗瓦行动适用于这些问题是插值到不断变化的家庭。打个比方，通过伽罗瓦动作的插值可以被认为是将一根弦穿过一组珠子。这个项目关注这些族中的“简并性”或“奇点”，类似于一个位于弦的股之间的收敛点的结。这个项目的目标不仅是“解开”这些简并以获得它们似乎掩盖的信息，而且还揭示了简并本身的额外数论信息。该项目的资金也将致力于支持宾夕法尼亚州西部的数学活动，例如将外部演讲者带到匹兹堡数论日，并鼓励学生在研究和推广方面的活动。至于学生研究，PI将建议研究生和本科生研究人员努力实现本项目的目标研究成果。至于推广，PI将招募和支持本科生在赠款资助的推广工作，以丰富小学和中学生的数学教育。伽罗瓦表示和模形式的p-adic变化的发展推动了现代代数数论的巨大进步。但当插值出现退化时，概念和工具缺乏或需要改进。这个项目的目的是解决和应用这些简并在各种设置使用同源工具。目标成果包括伽罗瓦表示和模形式之间联系的改进，p-基插值模形式的新概念在朗兰兹对应的衍生丰富的解释中的应用，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值进行评估来支持和更广泛的影响审查标准。

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.