On higher regulators of Picard modular surfaces

We prove the motivic classes in the motivic cohomology groups of Picard modular surfaces with non-trivial coefficients constructed in a paper of Loeffler\textendash Skinner\textendash Zerbes are in the motivic cohomology groups of the interior motives. Then we establish a relation between the motivic classes and non-critical values of the motivic $L$-functions associated to cuspidal automorphic representations of $\mathrm{G}\mathrm{U}(2,1)$, thus deducing non-triviality of the motivic classes and providing evidence for Beilinson’s conjectures.