Arithmetic Special Cycles and Derivatives of L-functions

摘要斯蒂芬·库德拉，马里兰大学，9970506，这个项目涉及到构造与某些下村簇上的代数圈有关的模生成级数。这种生成级数应该是模形式的Q-展开，这一事实反映了这些变种的代数子簇分布中的某种隐藏的对称性。此外，在某些情况下，存在一种更精细的生成级数类型，其系数位于Shimura簇的积分模型的算术Chow群中。这种级数似乎与Siegel-Eisenstein级数的导数有关，从而又与自同构的L函数的中心导数有关。因此，人们希望最终能得到格罗斯-扎吉尔公式和伯奇-斯温纳顿-戴尔猜想的高维类似物的信息。现代数论几何中现在出现的强大技术使处理以前以求解整数方程为中心的棘手问题成为可能。已发现与方程有关的图的几何性质与该方程的整数解之间有很强的联系。例如，著名的莫德尔猜想，由Faltings在1983年证明，它断言一个方程，其图形是一个至少有两个洞的曲面，只能有有限数量的整数解。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，而“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.)