Arithmetic Cycles, Eisenstein Series, Automorphic L-Functions, and Complex Multiplication

摘要：题目：算术循环、爱森斯坦级数、自同构l函数与复乘法。本课题研究了若干shimur变种上算术循环的高度对的生成函数与椭圆模和西格尔模爱森斯坦级数在某些临界点上的laurent展开的第二项之间的联系。生成函数可以看作是theta函数的算术类似物，并且可以用来定义经典theta对应的算术类似物，现在将某些类型的模形式用于算术Chow群的元素。一个应用是证明著名的Gross-Zagier公式在不改变l函数基底的情况下的厌恶性。人们也希望最终能够获得关于Gross-Zagier公式和birch - swinnerton - dyer猜想的高维类比的信息。本课题的另一部分是利用两条曲线的算法来研究密码学，这在电子通信中有非常实际的应用。在20世纪后期，“数论”几何的发展取得了重大进展，其中增加了一个额外的维度来携带涉及几何和素数之间相互作用的信息。在这样的空间中，我们可以给点加上一个称为高度的数字，这是对其“算术复杂度”的度量。更一般地说，高度可以定义为高维物体，例如曲面上的曲线。本项目研究这些高度之间的组合关系，反映了“数论”几何空间所携带的隐藏结构。电子通信（如信用卡处理）中最重要的部分之一是安全，这是通过加密技术来实现的。自80年代末以来，人们发现数论可以用于获得高度安全的密码系统。

DMS-0245406Yang, TonghaiAbstract:Title: Arithmetic Cycles, Eisenstein Series, Automorphic L-Functions, and Complex MultiplicationThis project involves the connections between generating functionsfor height pairings of arithmetic cycles on certain Shimuravarieties, on the one hand, and second terms in the Laurentexpansions of elliptic modular and Siegel modular Eisensteinseries at certain critical points, on the other. The generatingfunctions can be viewed as arithmetic analogues of theta functionsand can be use to define arithmetic analogues of the classicaltheta correspondence, now taking certain types of modular forms toelements of arithmetic Chow groups. One application is to prove aversion of the celebrated Gross-Zagier formula without base changethe L-function. It is also hoped that one mayultimately obtain information about higher dimensional analoguesof the Gross-Zagier formula and the Birch-Swinnerton-Dyerconjecture. Another part of the project is to use arithmetic ofgenus two curves to study cryptography, which has very practicalapplication in electronic communication.In the later part of the 20th century significant advances were made in developing a `number theoretic' geometry, in which an additional dimension is added to carry information involving the interaction between the geometry and prime numbers. To a point on such a space, one can attach a number call its height, which is a measure of its `arithmetic complexity'. More generally, heights can be defined for higher dimensional objects, curves on surfaces, for example. The present project studies combinatorial relations among such heights, which reflect hidden structure carried by the spaces of `numbertheoretic' geometry. One of the most important part in electroniccommunication such as credit card processing is security, which isdone by means of cryptography. Since late 80's, it is found thatnumber theory can be applied to obtain highly secure cryptographicsystem.