# Number Theory Seminar

Local densities of hermitian lattices are interesting counting invariants. Via local Whittaker functions, they are useful for understanding Fourier coefficients of Siegel Eisenstein series (with s-variable). For a p-adic lattice L, the constant local density Den(L) (with respect to a self-dual lattice) is the number of self-dual lattices containing L inside the hermitian space. Surprisingly, the Kudla—Rapoport conjecture (proved by Li—Zhang) says the first derived local density \partialDen(L) can be computed as intersection numbers of special cycles on local hermitian Shimura varieties with good reduction, at least when Den(L)=0. We will formulate Kudla—Rapoport type formulas for local densities with respect to more general lattices, where new phenomena occur. We will establish these formulas, assuming certain Tate conjectures for 1-cycles which is verified in many cases. Such formulas could be used to establish arithmetic Seigel—Weil formulas with mild ramifications, hence relate L-functions to algebraic cycles. The talk is based on joint work with Sungyoon Cho and Qiao He.