Norm-compatible systems of cohomology classes for operatorname{GU}(2,2)

We describe work of Faltings on the construction of \'etale cohomology classes associated to symplectic Shimura varieties and show that they satisfy certain trace compatibilities similar to the ones of Siegel units in the modular curve case. Starting from those, we construct a two variable family of trace-compatible classes in the cohomology of a unitary Shimura variety.