Abstract：In the nineties, Kudla formulated a conjecture relating central derivative of certain Eisenstein series to arithmetic intersection numbers of special cycles on Shimura varieties. Later Kudla and Rapoport formulated a local version of the conjecture which compares intersection numbers of special cycles on the unitary Rapoport Zink spaces over an inert prime of an imaginary quadratic field with derivatives of local density of hermitian forms. In this talk, I will review Kudla-Rapoport conjecture and its global motivation. Then I will talk about an attempt to formulate a Kudla-Rapoport type of conjecture over the ramified primes. In low rank cases, this new conjecture can be proved. This is a joint work in progress with Qiao He and Tonghai Yang.
