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

TL;DR

This paper explores the construction of compatible cohomology classes for unitary Shimura varieties, extending Faltings' work on symplectic varieties and establishing trace compatibility akin to modular units.

Contribution

It introduces a new two-variable family of trace-compatible cohomology classes for unitary Shimura varieties, building on Faltings' framework.

Findings

Construction of trace-compatible classes in cohomology.

Extension of Faltings' methods to unitary Shimura varieties.

Establishment of trace compatibility similar to modular units.

Abstract

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.