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 GU(2, 1) for an imaginary quadratic extension E/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. Contents 1. Introduction 1 2. Higher Chow classes on Picard modular surfaces 9 3. Rankin-Selberg method 20 4. Hecke operators and the regulator pairing 33 5. Kronecker limit formula 40 6. Prime number theorem 46 7. Local theory 47 8. Regulator computation 53 References 61

An Euler system for GU(2, 1)

“…We refer the reader to §8 for the definition of the Shimura variety Y G (K G ), and the relative Chow motive D a,b {r, s} over it. In the case (a, b, r, s) = (0, 0, 0, 0), this motive is simply the trivial motive E(0), and our classes coincide with those considered in [PS18]; in particular, the main result of op.cit. shows that the images of these classes under the Deligne-Beilinson regulator map, paired with suitable real-analytic differential forms on Y G (K G )(C), are related to the values L (π, 0) for cuspidal automorphic representations π of G(A).…”

“…This shows that our motivic cohomology classes are non-zero in this trivial-coefficient case. (We expect that a complex regulator formula similar to [PS18] should also hold for more general coefficient systems, but we shall not treat this problem here. )…”

Higher regulators of Siegel sixfolds and non-critical values of Spin $L$-functions

“…However the main problem of this approach is that there is no control on the Hodge type of the compactly supported differential forms that approximate ω Ψ . Moreover, it turned out that a similar gap appears in other works on complex regulators in different settings, such as [29], [33], [34], [41].…”

“…Proof. This is the second statement of [41,Corollary 5.6], where ν 1 is taken to be the trivial character.…”

On Higher regulators of Siegel varieties

“…Kronecker limit formula. Let us recall, following [31], Kronecker limit formula, relating the logarithm of the absolute value of modular units to classical Eisenstein series. 4.4.1.…”