Shimura Varieties and Galois Modules

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The principal investigator is working on the following two problems:(A) He is attempting to describe integral models for Shimura varieties at primes of non-smooth reduction. In particular, he studies ``local models" for PEL Shimura varieties and their relation with affine flag varieties for infinite dimensional groups, with symmetric spaces and with deformation spaces of Galois representations.The motivation is to obtain information that can be used in the calculation of the Hasse-Weil zeta function of these varieties and in other arithmetic applications. (B) He is studying the representations that appear in the cohomology of arithmetic varieties with a group action.In particular, he continues his work on developing fixed point formulas for calculating invariants of such (integral) representations usingthe theory of cubic structures.The investigator's research is in the field of arithmetic algebraic geometry, a subject that blends two of the oldest areas of mathematics:the geometry of figures that can be defined by the simplest equations, namely polynomials, and the study of numbers. This combination has proved extraordinarily fruitful - having solved problems that withstood generations (such as ``Fermat's last theorem"). The investigator's work mainly concentrates on the study of certain polynomial equations that have many symmetries. There are connections with physics, the construction of error correcting codes and cryptography.