Arithmetic Special Cycles and Derivatives of L-functions

马里兰州9970506本项目涉及与某些志村簇上的代数圈相关的模生成级数的构造。这样的生成级数应该是模形式的q-展开式这一事实反映了这些簇的代数子簇的分布中的某种隐藏的对称性。此外，在某些情况下，有一个更微妙的类型生成系列的系数在于算术周群的一个完整的模型的志村品种。这样的级数似乎与Siegel-Eisenstein级数的导数有关，因此反过来又与自守L-函数的中心导数有关。因此，人们希望最终可以获得关于Gross-Zagier公式和Birch-Swinnerton Dyer猜想的高维类似物的信息。强大的技术，现在出现在现代数论几何已经有可能管理以前棘手的问题集中求解方程的整数。在与方程相关联的图的几何性质和该方程的整数解之间已经发现了强联系。 例如，著名的莫德尔猜想，证明了法尔明斯在1983年，断言一个方程的图是一个表面至少有两个洞可以有一个有限数量的整数解。目前的项目涉及这种联系的更复杂的版本，其中方程及其图形需要额外的维度，并且“整数解”实际上可能是其他类型的几何对象（曲线，曲面等）。

Abstract Stephen Kudla University of Maryland 9970506 This project involves the construction of modular generating series associated to algebraic cycles on certain Shimura varieties. The fact that such generating series should be the q-expansions of modular forms reflects some sort of hidden symmetry in the distribution of algebraic subvarieties of these varieties. In addition, in certain cases, there is a more delicate type of generating series whose coefficients lie in the arithmetic Chow groups of an integral model of a Shimura variety. Such series appear to be related to the derivatives of Siegel-Eisenstein series and hence, in turn, to the central derivatives of automorphic L-functions. Thus, it is hoped that one may ultimately obtain information about higher dimensional analogues of the Gross-Zagier formula and the Birch-Swinnerton Dyer conjecture. The powerful techniques which are now emerging in modern number theoretic geometry have made it possible to manage previously intractable problems centered about solving equations with integers. Strong connections have been found between the geometric properties of graphs associated with an equation and the integer solutions to that equation. For example, the celebrated Mordell conjecture, proved by Faltings in 1983, asserts an equation whose graph is a surface with at least two holes can have only a finite number of integer solutions. The present project involves more complicated versions of this connection in which the equations and their graphs require extra dimensions and the `integer solutions' may actually be other kinds of geometric objects (curves, surfaces, etc.)