CAREER: Arithmetic and Geometry of Pure and Mixed Shimura Varieties

A central theme in modern number theory is the conjectural relations between classes of automorphic representations and Galois representations, with notions of algebraicity on both sides, in the context of Langlands program and its p-adic and p-torsion analogues. To realize such conjectural relations at all, most known methods so far involve the use of the cohomology of certain algebraic varieties and their models over the integers; namely, the so-called (pure) Shimura varieties, the Kuga families over them (generalizing the Kuga--Sato varieties over modular curves) which are special cases of the so-called mixed Shimura varieties, their compactifications, and good models of these over the integers. The research projects in this proposal aim at making further progress in understanding such fundamentally important geometric objects, through extensive studies along directions both old and new, and at developing their new arithmetic applications.

Number theory and geometry are the two oldest branches of mathematics, and combined applications of them (such as error-correcting codes) have become indispensable in modern daily life (involving, for example, telecommunication and data storage). The pure and mixed Shimura varieties are important geometric objects relating algebra, analysis, and geometry in natural yet mysterious ways, and advances in their theory have contributed to many exciting recent developments in number theory. The education projects in this proposal aim at creating a vertically integrated learning environment for number theory and arithmetic geometry at the University of Minnesota, from which students at all levels can benefit, including supports for outreach activities, summer learning projects, and the development of courses and learning seminars.