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。PI及其同事最近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范围。这导致了有用的有限定理的推广，并提出了PI提出要解决的关于函数域上的线性代数群的新问题。PI将继续研究非阿基米德几何中的下降理论，以及伽罗瓦变形理论中涉及约化群、刚性解析几何和p进霍奇理论的一些问题。最后，PI将研究宇称障碍的更深层性质，包括在全局函数场上椭圆曲线族中的莫德尔-韦尔秩的应用，并为研究具有几何风味的数论问题的博士生提供建议。尽管研究数论的动机大多来自纯数学，但重要的实际应用仍在不断涌现。例如，安全的互联网通信主要依赖于出于理论原因而发明的几何起源的数论技术。这一建议解决了数论几何方面积极发展的几个领域的问题，并将导致解决具体问题的新的理论方法，并为未来的进展奠定基础。PI还建议继续监督有才华的高中生的数论研究，并在高中、本科和研究生阶段的学生群体中进行许多数学领域的演讲。

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.