Arithmetic Siegel-Weil Formulas and Arithmetic Theta Lifting

这个项目一方面涉及某些Shimura变种上算术圈的高度对的母函数，另一方面涉及椭圆模和Siegel模Eisenstein级数在某些临界点的Laurent展开式中的第二项之间的联系。生成函数可以被视为theta函数的算术模拟，并且可以用来定义经典theta对应的算术模拟，现在对算术Chow群的元素采用某些类型的模形式。一个主要的目标是证明Rallis关于这些算术提升力的内积公式的类似，现在将涉及L函数的导数。因此，人们希望最终能够获得有关格罗斯-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。在20世纪后期，在发展“数论”几何学方面取得了重大进展，在该几何学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。在这样一个空间上的一个点上，人们可以把一个数字附加到它的高度，这是对它的‘算术复杂性’的衡量。更一般地，可以为更高维的对象定义高度，例如曲面上的曲线。本项目研究了这些高度之间的组合关系，这些关系反映了“数论”几何空间所承载的隐藏结构。

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.