Kim 9701489 This award provides funds for M. Kim to continue his past research on the Diophantine geometry of curves over function fields of arbitrary characteristic. In elementary language, this is the study of equations of type f(x,y,t)=0 (1) viewed as a two-variable polynomial equation whose coefficients are functions of t to which we seek solutions (x,y)=(p(t), q(t)), that is, pairs of polynomials (or rational functions) which satisfy f(p(t),q(t),t)=0 as a function of t. In particular, he will continue his previous work on `effective' Mordell conjectures over function fields, which allows one to find all solutions to (1) (for f's of high genus) by giving a priori bounds on the `height' (which in this case is the degree) of solution pairs (p(t),q(t)). A result of this sort over number fields is the long-term eventual goal of this research This is an outstanding problem in Diophantine geometry whose resolution would provide a crude algorithm for finding all rational solutions to Diophantine equations in two variables over the rational numbers. The specific projects go in three closely related directions: (1) finding more refined geometric height inequalities in arbitrary characteristic to serve as efficient input into algorithmic search for solutions; (2) embedding the PI's previous work into the study of curves on surfaces of general type in the spirit of Bogomolov's theorem on boundedness of curves of bounded genus, and that of his joint work with Shepherd-Barron; (3) approaching geometric height inequalities from a more explicitly algorithmic viewpoint in the hope of finding techniques which would transfer more readily to number fields than the existing ones. 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 corre cting codes. Such codes are essential for both modern computers (hard disks) and compact disks.
