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

点击翻译按钮获取中文摘要

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.