Zeta dimension formula for Picard modular cusp forms of neat natural congruence subgroups
Let \(K\) be an imaginary quadratic field, and let \(\mathfrak O_K\) be its ring of integers. If \(\Gamma_K = \mathbb U ((2,1), \mathfrak O_K)\) is the full Picard modular group of \(K\), then for each \(m \geq 3\) the natural congruence subgroup \(\Gamma_K (m)\) acts freely on the two-dimensional complex unit ball \(\mathbb B = \{ (z_1, z_2) \in \mathbb C^2 \mid |z_1|^2 + |z_2|^2 < 1 \}\). In this paper, the author derives an explicit polynomial formula for the dimensions of the spaces of cusp forms of weight \(n \geq 2\) for such a congruence subgroup \(\Gamma_K (m)\). The coefficients of these polynomials in the natural variables \(m\) and \(n\) are expressed by some Bernoulli numbers of the Dirichlet characters of \(K\) and by values of Euler factors of zeta functions. The proof is based on detailed knowledge about classification of Picard modular surfaces and combines algebraic geometric methods with number theoretic ones.
scientific article; zbMATH DE number 1416658
scientific article