Zeta dimension formula for picard modular cusp forms of neat natural congruence subgroups

Abstract

Let $$\Gamma _K = \mathbb{U}((2,1),\mathfrak{D}_K )$$ be the full Picard modular group of the imaginary quadratic number fieldK. For all natural congruence subgroups Γk (m), m ≥ 3, acting freely on the two-dimensional complex unit ball, we prove an explicit polynomial formula for the dimensions of spaces of cusp forms of weightn ≥ 2. The coefficients of these polynomials in the natural variablesm, n are expressed by higher third and first Bernoulli numbers of the Dirichlet character χk of K and by values of Euler factors of the Riemann Zeta function and such factors of theL-series of χkat 2 or 3, respectively. The proof is based on detailed knowledge about classification of Picard modular surfaces. It combines algebraic geometric methods (Riemann-Roch, Vanishing- and Proportionality Theorem, curvature, structure of algebraic groups) with modern and classical number theoretic ones (representation densities, Tamagawa measure, strong approximation, functional equation forL-series).