Arithmetic cycles on Shimura varieties

To construct algebraic cycles on algebraic varieties is a major challenge for which no general method exists. For Shimura varieties representing moduli problems of abelian varieties, it is possible to exhibit such cycles by imposing on the abelian varieties special structures, e.g. additional endomorphisms. This project studies such cycles  in the arithmetic context. We will investigate  the intersection numbers of such special arithmetic cycles, and provide further evidence for Kudla's suggestion that generating series formed from these intersection numbers are modular forms.