Collaborative Research / FRG: Arakelov Theory and Modular Forms

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FRG collaborative Award with lead DMS-0354353 of Ono, Zhang and Kudla with Co-PI Yang. This project involves arithmetic geometry and number theory andfocuses on a systematic study of cycles on Shimura varieties and applications. The fascination of diophantine problems -- the study of whole numbersolutions of polynomial equations-- goes back to ancient times.The mathematical techniques developed in the last 50 year toattack such questions have lead to significant advances in our knowledgeof this subject, for example the proof of Fermat's last theorem.These same tools have, meanwhile, proved to be of great importancein cryptography, the construction of new algorithms for computer scienceand new error correcting codes for electronics.Shimura varieties are the geometric objectsassociated 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 ofcycles on Shimura varieties, with an emphasis on the interactionof their heights, arithmetic intersections and density propertieswith modular forms and special values of L-functions and theirderivatives. Applications will be made to Gauss's class numberproblem, equidistribution problems for cycles on Shimuravarieties, and the Andre-Oort conjecture, and the Tate and Bloch-Beilinsonconjectures. This collaborative project will takeadvantage of recent developments including nonvanishing propertiesof 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 relationsbetween the height pairings, periods, and algebraic cycles and thederivatives of L-functions.