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.
