Mathematical Sciences: Shafarevich-Tate Groups

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This award supports the research in arithmetic algebraic geometry of Professor William McCallum of the University of Arizona. Dr. McCallum's project is to study the Shafarevich-Tate group of elliptic curves and Abelian varieties defined over number fields in order to verify a number of conjectures of Kolyvagin. The elliptic curves under consideration will be modular, while the Abelian varieties to be studied will all be Jacobian varieties of Fermat curves. Arithmetic algebraic geometry is 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.