A class number formula for Picard modular surfaces

Abstract

We investigate arithmetic aspects of the middle degree cohomology of compactified Picard modular surfaces $X$ attached to the unitary similitude group $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$. 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.

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Cited by in corpus (2)

• An Euler system for GU(2, 1)
• Tempered currents and Deligne cohomology of Shimura varieties, with an application to ${\mathrm{GSp}}_6$