This grant supports the research of Professor N. Katz as principal investigator and Drs. H. Darmon and F. Rodriquez- Villegas as post-doctoral associates. Professor Katz will work on the l-adic cohomology of varieties over finite fields, the l-adic theory of exponential sums and the arithmetic theory of differential equations. Dr. Darmon will study relations between the arithmetic of schemes and the analytic properties of l-functions. Dr. Rodriquez-Villegas will study the arithmetic properties of special values of l-functions of CM-fields. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.
