FRG collaborative Award with lead DMS-0354353 of Ono, Zhang and Kudla with Co-PI Yang.
This project involves arithmetic geometry and number theory and focuses on a systematic study of cycles on Shimura varieties and applications. The fascination of diophantine problems -- the study of whole number solutions of polynomial equations-- goes back to ancient times. The mathematical techniques developed in the last 50 year to attack such questions have lead to significant advances in our knowledge of this subject, for example the proof of Fermat's last theorem. These same tools have, meanwhile, proved to be of great importance in cryptography, the construction of new algorithms for computer science and new error correcting codes for electronics. Shimura varieties are the geometric objects associated to systems of diophantine equations with a great degree of symmetry. Their diophantine properties have deep connection with many important parts of mathematics.
An extensive study will be made of the arithmetic geometry of cycles on Shimura varieties, with an emphasis on the interaction of their heights, arithmetic intersections and density properties with modular forms and special values of L-functions and their derivatives. Applications will be made to Gauss's class number problem, equidistribution problems for cycles on Shimura varieties, and the Andre-Oort conjecture, and the Tate and Bloch-Beilinson conjectures. This collaborative project will take advantage of recent developments including nonvanishing properties of Fourier coefficients of modular forms, the theory of Borcherds forms and their connections with theta functions, and integral representations of Langlands L-functions. A long range goal of the project is to establish relations between the height pairings, periods, and algebraic cycles and the derivatives of L-functions.
