Shrenik Shah (Columbia University)
A class number formula for Picard modular surfaces

Abstract: We investigate arithmetic aspects of the cohomology of the smooth compactified Picard modular surfaces $X$ attached to the unitary group $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$. We begin by constructing suitable elements in the motivic cohomology $H_\mathcal{M}3(X,\mathbf{Q}(2))$. We then compute their regulator as an element of Deligne cohomology by interpreting this map via the pairing of these classes against automorphic differential forms, and show that the regulator is non-vanishing when predicted. As a consequence, we obtain a special value formula, akin to a class number formula, involving a non-critical $L$-value of the degree 6 Standard L-function, a Whittaker period, and the regulator. Our investigation provides support for Beilinson's regulator conjecture in this setting. One interesting aspect of this work is that we must account for endoscopic forms via a period, which is predicted by the conjecture. This is joint work with Aaron Pollack.