This award supports the research in arithmetic algebraic geometry of Professor Jeremy Teitelbaum of the University of Illinois at Chicago. Dr. Teitelbaum will pursue his studies of the geometry and arithmetic of the p-adic upper half plane, guided by the Exceptional Zero Conjecture, which relates the values of certain functions, namely p-adic L-functions, to the analytic geometry of certain curves. In addition, he will continue his study of the computational complexity of algorithms from algebraic geometry. 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.
