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

DMS-0245406杨通海摘要：题目：算术循环，爱森斯坦级数， 自守L-函数和复数乘法这个项目涉及的生成函数之间的连接为高度配对的算术循环在某些Shimuravarieties，一方面，第二项在劳伦展开的椭圆模和西格尔模Eisensteinseries在某些临界点，另一方面。生成函数可以被看作是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公式的不换基L-函数的厌恶性。 人们也希望最终能获得关于Gross-Zagier公式和Birch-Swinnerton-Dyer猜想的高维类比的信息。该计划的另一部分是利用亏格两条曲线的算法来研究密码学，这在电子通信中有着非常实际的应用。在世纪后期，在发展“数论”几何方面取得了重大进展，在几何中增加了一个额外的维度来携带涉及几何与素数之间相互作用的信息。对于这样一个空间上的一个点，我们可以附加一个称为它的高度的数字，这是它的“算术复杂性”的度量。更一般地，可以为更高维度的对象（例如，曲面上的曲线）定义高度。本项目研究这些高度之间的组合关系，这反映了“数论”几何空间所携带的隐藏结构。在电子通信如信用卡处理中，最重要的部分之一是安全性，这是通过密码学来实现的。80年代后期以来，人们发现数论可以用来构造高安全性的密码体制。

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.