Algebra and Geometry Seminar

We study a collection of special cycles on unitary Shimura varieties that are higher-dimensio​nal analogues of Heegner points on modular curves, and describe these cycles adelically. Using Bruhat--Tits theory for unitary groups, we prove that they satisfy certain rigidity relations (known as distribution relations), thus allowing to recover arithmetic information from cohomological data (cycle class images under p-adic Abel--Jacobi maps). We then state certain equidistributio​n conjectures for the specialization of these cycles to the special fiber of the (canonical) integral models of the ambient Shimura varieties related to non-vanishing of the cohomology classes (the latter formulated in terms of superspecial abelian varieties and mass formulas for unitary groups). As arithmetic applications of the distribution relations and the non-vanishing conjectures, we prove novel results towards the Bloch--Kato--Be​ilinson conjectures for ranks of groups of rational cycles in terms of analytic ranks of automorphic L-functions.