On certain families of special cycles on Shimura varieties

Abstract

In this two-part thesis, we study certain families of special cycles on Shimura varieties, which have interesting arithmetic applications. In the first part, extending the ideas of Darmon and Rotger, we construct a p-adic family of Hirzebruch-Zagier cycles on Shimura threefolds obtained from products of modular curves and Hilbert modular surfaces. This family gives rise to a big cohomology class residing in the Galois cohomology of a certain Lambda-adic Galois representation. We then establish a regulator formula for the cycles, which allows us to relate the big cohomology class to a twisted triple product p-adic L-function. As an application, we establish new instances of the equivariant BSD-conjecture in rank 0 and study the arithmetic of rational elliptic curves over quintic fields. In the second part, we follow the ideas of Loefller, Skinner and Zerbes to set up and conduct local automorphic computations on the algebraic group GSp4 x GL2, establishing technical results that should lead to the norm relations of an Euler system for GSp4 x GL2.