Title: Arithmetic special cycles for unitary groups

Abstract: In this talk I will report on some recent work in progress with M. Rapoport. In a long series of papers with R. and Tonghai Yang, we have discussed the connections between arithmetic geometry and modular forms arising from algebraic cycles on Shimura varieties for orthogonal groups O(n,2) over Q. A drawback of these examples is the fact that these Shimura varieties are only of PEL type for small values of n, although they are of Hodge type in general. The accessible examples are then limited by the fact that the calculations of arithmetic intersection numbers are all based on the modular interpretation. On the other hand, in developing the arithmetic theory, we are frequently guided by the geometric results about cycles, originating in Hirzebruch-Zagier and extended by KM. In the current project, our aim is to begin the arithmetic study of the cycles for U(r,s) type Shimura varieties.