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.
