Arithmetic Siegel-Weil Formulas and Arithmetic Theta Lifting

本课题研究了某些Shimura变数上算术循环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函数的算术类似物，可以用来定义经典对应的算术类似物，现在对算术周氏群的元素采取某些类型的模形式。我们的主要目标是证明Rallis的内积公式对于这些算术上的提升的相似之处，这将涉及到l函数的导数。因此，人们希望最终能够获得关于Gross-Zagier公式和Birch-Swinnerton-Dyer猜想的高维类似物的信息。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，这些高度反映了“数论”几何空间所携带的隐藏结构。

This project involves the connections between generating functions for height pairings of arithmetic cycles on certain Shimura varieties, on the one hand, and second terms in the Laurent expansions of elliptic modular and Siegel modular Eisenstein series at certain critical points, on the other. The generating functions can be viewed as arithmetic analogues of theta functions and can be use to define arithmetic analogues of the classical theta correspondence, now taking certain types of modular forms to elements of arithmetic Chow groups. A main goal is to prove analogues of Rallis's inner product formula for these arithmetic theta lifts, which will now involve derivatives of 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. 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 `number theoretic' geometry.