FRG: Arakelov Theory and Modular Forms

FRG与小野、张、库德拉的首席DMS-0354353合作获奖，杨协皮。本项目涉及算术几何和数论，重点是对Shimura变种及其应用的循环进行系统研究。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对这门学科的知识有了显著的进步，例如费马最后定理的证明。同时，这些同样的工具在密码学、计算机科学的新算法的构造和电子学的新纠错码中都被证明是非常重要的。下村变种是与具有很大对称性的丢番图方程组有关的几何对象。它们的丢番图性质与数学的许多重要部分有着密切的联系。本文将广泛地研究Shimura簇上圈的算术几何，重点讨论它们的高度、算术交集、稠密性与L函数及其导数的模形式和特殊值之间的相互作用。我们将应用于Gauss的类数问题，Shimuravies上圈的均匀分布问题，Andre-Oort猜想，以及Tate和Bloch-Beilinsson猜想。这一合作项目将利用最近的发展，包括模形式的傅里叶系数的非零性，Borcher形式的理论及其与theta函数的联系，以及朗兰兹L函数的积分表示。该项目的一个长期目标是建立高度对、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.