On classes in the motivic cohomology of certain Shimura varieties

The main theme of this thesis is on the push-forward construction of motivic cohomology classes for Shimura varieties. This strategy was successfully employed in work of Lei-Loeffler-Zerbes et al to construct new Euler systems for Galois representations attached to certain cohomological automorphic forms, which have been used to prove new cases of the Bloch-Kato conjecture. In this thesis, we describe two new push-forward constructions for Shimura varieties associated to the symplectic group GSp(6) and the unitary group GU(2,2), and their distribution relations. First, we describe the joint work with Joaquin Rodrigues Jacinto on the construction of classes in the seventh cohomology group of the Shimura variety for GSp(6); these classes have coefficients in a local system associated to an irreducible algebraic representation of GSp(6) of arbitrary weight. The classes are defined as push-forward of elements in the cohomology of a triple product of modular curves. We prove a trace compatibility result for these classes and use it to deduce Euler system norm relations in the cyclotomic tower at any rational prime p. Secondly, we explain the construction of classes in the fifth motivic cohomology group of the Shimura variety for GU(2,2). They are obtained as the push-forward of GSp(4)-Eisenstein classes along the Gysin morphisms of a closed immersion of the Shimura variety for GSp(4) inside the one for GU(2,2). By perturbing the aforementioned immersion, we construct a two variable family of push-forward classes that satisfies certain norm relations. To derive these, we first prove, more generally, some distribution relations for the GSp(2g)-Eisenstein classes and then translate them into those for the push-forward classes