A class number formula for Picard modular surfaces

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