Mathematical Sciences: The Arithmetic of Shimura Varieties

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This award supports the research of Professor James Milne to work in algebraic geometry. It has been known for several decades that there is a close and fruitful relation between the arithmetic of abelian varieties and their degenerations, on the one hand, and the arithmetic of Siegel modular varieties and their compactifications on the other hand. Recently, Professor Milne has found a similar relation between abelian motives and Shimura varieties of abelian type. It is proposed to study the degeneration of abelian type. It is proposed to study the degeneration of abelian motives and exploit the relation in order to obtain information about the compactifications of Shimura varieties over number fields This research uses the techniques of algebraic geometry, 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.