Mathematical Sciences: Arithmetic Models for Shimura Varieties, L-Functions and Cohomology Groups as Integral Representations

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9623269 Pappas The PI will continue his work on the following two problems: A. Study the geometry and the local properties of arithmetic models of Shimura varieties over primes of non-smooth reduction, by using ideas from the theory of equivariant embeddings of homogeneous spaces. B. Study the integral representations that appear in the cohomology of arithmetic varieties with finite group actions. This is part of a program for extending the theory of Galois module structure of rings of integers of A. Fr;3hlich and M. Taylor to higher dimensional arithmetic varieties. The author proposes to investigate relations of invariants describing these representations with data obtained from L-functions. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.