Number Theory Seminar

I will speak about an application of the regularized Siegel-Weil formula to intersection numbers of diagonal and special cycles on a unitary Shimura variety. The central apparatus is the Weil representation and four reductive groups forming see-saw dual pairs. For a specific choice of an adelic Schwartz function derived from a construction of Kudla-Millson, the weak second-term Siegel-Weil formula describing the residue of a regularized theta integral can be strengthened to a precise identity. Thus one obtains new cases of the general work of Kudla-Millson on modularity of generating series of special cycles. The intersection of this series with the diagonal cycle is identified as the Fourier expansion of the constant term in the Laurent series of a Siegel Eisenstein series at the edge point of convergence.