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We investigate arithmetic aspects of the middle degree cohomology ofcompactified Picard modular surfaces $X$ attached to the unitary similitudegroup $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$.We construct new Beilinson--Flach classes on $X$ and compute their Archimedeanregulator. We obtain a special value formula involving a non-critical $L$-valueof the degree six standard $L$-function, a Whittaker period, and the regulator.This provides evidence for Beilinson's conjecture in this setting.

A class number formula for Picard modular surfaces