Collaborative Research / FRG: Arakelov Theory and Modular Forms

FRG与Ono、Zhang和Kudla的DMS-0354353负责人以及Co-PI Yang合作奖。该项目涉及算术几何和数论，重点是对志村簇上的圈及其应用进行系统的研究。丢番图问题的魅力--研究多项式方程的整数解--可以追溯到古代。在过去的50年里，为解决这类问题而发展起来的数学技术使我们在这方面的知识有了重大的进步，例如费马最后定理的证明。同时，这些同样的工具在密码学中也被证明是非常重要的，Shimura簇是与具有高度对称性的丢番图方程组相关联的几何对象。它们的丢番图性质与数学的许多重要部分有着深刻的联系，本文将对Shimura簇上的圈的算术几何进行广泛的研究，重点是它们的高度、算术交和密度性质与L-函数及其导数的模形式和特殊值的相互作用。应用程序将高斯的类numberproblem，equidistribution问题的周期Shimuravariteties，和安德烈-奥尔特猜想，和泰特和布洛赫-贝林sonatrices。 这个合作项目将利用最近的发展，包括非零性质的傅立叶系数的模形式，理论的Borcherds形式及其与theta函数的连接，和积分表示的朗兰兹L-功能。该项目的一个长期目标是建立高度配对，周期，代数圈和L-函数的导数之间的关系。

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.