**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.