Mathematical Sciences: Level Structures on Abelian Varieties

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This award supports the research on level structures for Abelian varieties of Professor Peter Norman of the University of Massachusetts. Dr. Norman's work will involve the problems of lifting Abelian varieties from positive characteristic to zero characteristic, as well as researches on Hilbert-Blumenthal varieties and Dieudonne theory. This is research in the field of arithmetic algebraic geometry, a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.