Shimura Varieties, Galois Modules and the Determinant of Cohomology

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The 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 tries to formulate exact predictions about the non-smooth reduction of a general Shimura variety. 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. He will continue his work on describing relations between invariants describing such representations and L-functions and develop a theory of ``non-abelian" cubic structures for the determinant of cohomology of general bundles over curves.This is research 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 provedextraordinarily fruitful - having recently solved problems that withstood generations (such as ``Fermat's last theorem"). The investigator's work 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.