Abstract

The original class number formula of Dirichlet connected the residue of the Dedekind zeta function of a number field at s=1 to various arithmetic invariants of the number field, including a transcendental quantity called the regulator, which measures the covolume of a lattice generated by logarithms of units. Beilinson formulated a conjectural generalization of this formula, again connecting a regulator, defined in terms of “higher” units, to the special value of a “motivic” L-function. We study a particular case of this conjecture, namely for the motive of the middle degree cohomology of the smooth compactified Picard modular surfaces X attached to the unitary group GU(2,1) for an imaginary quadratic extension E/Q. We will concretely describe a construction of suitable elements in the motivic cohomology H_M^3(X,Q(2)). We then explain how to 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 higher “class number formula” involving a non-critical L-value of the degree 6 Standard L-function, a Whittaker period, and the regulator. 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.