Abstract: The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport-Zink spaces and the derivatives of local representation densities of hermitian forms. It is a key local ingredient to establish the arithmetic Siegel-Weil formula, relating the height of generating series of special cycles on Shimura varieties to the derivative of Eisenstein series. We discuss a proof of this conjecture and global applications. This is joint work with Wei Zhang.
Finiteness of small eigenvalues of geometrically finite manifolds
Let M be a geometrically finite real rank one locally symmetric manifolds. We will take about the spectrum of the Laplace operator on M. By using the Lax-Philips inequality of the energy form, we will prove that the spectrum is finite in a critical interval which is given by the volume entropy.