The PI is attempting to describe integral models for Shimura varieties at primes of non-smooth reduction and study related p-adic spaces. In particular, he investigates the structure of the singularities of Shimura varieties at such primes using methods from the theory of algebraic loop groups and p-adic Hodge theory and he is developing a theory of ``local models" for these varieties. The motivation is to obtain information that can be used in the calculation of their Hasse-Weil zeta functions and in other number theoretic applications. He is also studying closely connected p-adic spaces such as deformation spaces of local Galois representations, Rapoport-Zink spaces and their p-adic cohomology. In addition he is studying the representations that appear in the cohomology of arithmetic varieties with a finite group action and is developing related Riemann-Roch type theorems. In the past, the PI has obtained formulae useful for calculating equivariant Euler characteristics of varieties over the integers with an abelian group action. He is extending his work to the case of non-abelian group actions and he will apply his techniques to deduce information about Galois modules obtained from covers of modular curves. He is also developing refined versions of Riemann-Roch type theorems that allow for calculations of torsion classes, and, in some cases, he is providing local or adelic descriptions of the isomorphisms underlying the Riemann-Roch identities.
This is research in the field of arithmetic algebraic geometry,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 problems that withstood generations (such as ``Fermat's last theorem"). The proposal mainly concentrates on the study of specific polynomial equations that have many symmetries. The general field has connections with physics, the construction of error correcting codes and cryptography.
