Arithmetic Geometry

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The award supports the principal investigator's research in arithmetic geometry. Arithmetic geometry is a branch of mathematics that studies integer solutions of polynomial equations and has played a central role in solving many outstanding problems in number theory, such as Fermat's Last Theorem and the Mordell conjecture concerning the number of rational points on a curve. The main objects of study in this research project are Shimura varieties, which are higher dimensional analogues of modular curves that parametrize families of Riemann surfaces. The study of Shimura varieties is at the interface of algebraic geometry, number theory, and representation theory and has broad applications to a number of far-reaching and influential conjectures.One of the main topics of this research project concerns the arithmetic of Shimura varieties. One part of the work aims to relate the Hasse-Weil zeta function of a certain class of Shimura varieties -- those of Hodge type -- to automorphic L-functions, establishing a conjecture of Langlands. Another aims to study the structure of special points on these varieties and to show that, in certain situations, every mod p isogeny class contains the reduction of a special point. Other directions in the project include the Grothendieck-Katz p-curvature conjecture, as well Diophantine methods for constructing algebraic cycles.