Arithmetic Geometry: Shimura Varieties, Galois Modules, and Iwasawa Theory

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This project concerns research in the field of arithmetic algebraic geometry. This is a subject that blends two of the oldest areas of mathematics: the geometry of shapes that can be described by the simplest equations, namely polynomials, and the study of numbers. This combination of disciplines has proved extraordinarily fruitful, having solved challenges (such as the recent proof of Fermat's last theorem) that had withstood generations of effort. The field has connections with physics, and has found important applications to the construction of error-correcting codes and cryptography. This research focuses on the study of specific polynomial equations that have many symmetries. This project aims to describe integral models for Shimura varieties at primes of non-smooth reduction and will study related p-adic spaces. In particular, the work continues investigation of the singularities of Shimura varieties of abelian type at such primes. The project aims to characterize these integral models via a suitable Neron extension property and, in the case of orthogonal Shimura varieties, explicitly study the local structure of their reductions. The project also intends to give a general construction and a group theoretic definition of integral models of certain Rapoport-Zink p-adic spaces and, in some cases, fully describe their special fibers. The project additionally studies the representations that appear in the cohomology of varieties over the integers with a finite group action and aims to develop very general fixed point formulae that can be used to calculate equivariant Euler characteristics. Finally, the project explores extending constructions of Iwasawa theory by employing K-theoretic methods; more specifically, obtaining information about the higher codimension primes of the Iwasawa algebra that lie on the support of an Iwasawa module.