Mathematical Sciences: Weights of Semisimple Lie Algebras Arising from the Cohomology of Algebraic Varieties

This grant supports the work of Professor Y. Zarhin to work in arithmetic geometry specifically problems in l-adic cohomology. He will study how the weights of an l-adic representation can restrict the Lie algebra representations arising from Galois actions on Abelian varieties. He will also study the eigenvalues of the Frobenius action on Abelian varieties over finite fields. He also intends to study complex Abelian varieties where the rank of the Hodge group is close to the dimension. 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.