Problems Over Local Fields and Function Fields

这个项目的重点是代数几何思想，可以用来提高我们对算术现象的理解。最近，PI等人通过对伪约化群的研究，扩展了函数域上的约化代数群理论的研究范围。这导致了有用的有限定理的推广，并提出了PI建议解决的与函数域上的线性代数群有关的新问题。PI将继续他在非阿基米德几何中的下降理论以及伽罗瓦变形理论中的一些问题的工作，这些问题涉及约化群、严格解析几何和p-进Hodge理论。最后，PI将研究奇偶障碍的更深层次的性质，包括在全局函数场上的椭圆曲线族中的Mordell-Weil列的应用，并建议博士生从事具有几何风格的数论问题。尽管学习数论的大部分动机来自于纯数学，但重要的实际应用继续涌现。例如，安全的互联网通信至关重要地依赖于几何起源的数论技术，这些技术是出于理论原因而发明的。这一建议解决了数论几何方面活跃发展的几个领域中的问题，并将导致解决具体问题的新理论方法，并为未来的进展奠定基础。国际学生联合会还建议继续监督有天赋的高中生进行数论研究，并向高中、本科生和研究生水平的学生群体讲授许多数学领域的课程。

This project focuses on algebro-geometric ideas that can be used to improve our understanding of arithmetic phenomena. The PI and co-workers have recently extended the scope of the theory of reductive algebraic groups over function fields via their work on pseudo-reductive groups. This has led to generalizations of useful finiteness theorems, and suggested new problems concerning linear algebraic groups over function fields that the PI proposes to solve. The PI will continue his work on descent theory in non-archimedean geometry, as well as on some problems in Galois deformation theory that involve reductive groups, rigid-analytic geometry, and p-adic Hodge theory. Finally, the PI will study deeper properties of a parity obstruction, including applications to Mordell-Weil ranks in families of elliptic curves over global function fields and advise PhD students working on number-theoretic problems with a geometric flavor.Although much of the motivation for the study of number theory comes from within pure mathematics, important practical applications continue to emerge. For example, secure Internet communication relies crucially on number-theoretic techniques of geometric origin that were invented for theoretical reasons. This proposal addresses questions in several areas of active development on the geometric side of number theory, and will lead to new theoretical methods that solve specific problems as well as lay the groundwork for future progress. The PI also proposes to continue supervising number theory research by talented high school students and giving talks across many areas of mathematics to groups of students at the high school, undergraduate, and graduate levels.