Invariants of some compactified Picard modular surfaces and applications (Q2629759)

Let \(K\) be an imaginary quadratic field with ring of integers \({O}_K\). Let \(\text{SU}(h,{O}_K)\) be the special unitary group associated to a Hermetian form \(h\) of signature \((2,1)\) on a \(3\)-dimensional vector space over \(K\). Let \(\Gamma_K = \text{SL}_3({O}_K) \cap \text{SU}(h,{O}_K)\) be the full Picard modular group and \(\Gamma_K(\mathfrak a)\) be the principal congruence subgroup for integral ideals \(\mathfrak a\) in \({O}_K\). The author computes the index \([\Gamma_K : \Gamma_K(\mathfrak a)]\) and determines when \(\Gamma_K(\mathfrak a)\) is neat. Once in the neat case, the compactification at cusps of the associated quotient spaces is given by smooth elliptic curves, and the paper computes Chern classes of the associated quotient spaces \(\Gamma_K(\mathfrak a) \backslash \mathbb B\). This is applied to determine when the smooth compactification of \(\Gamma_K(\mathfrak a) \backslash \mathbb B\) is a surface of general type, and dimension formulae for the space of cusp forms for \(\Gamma_K(\mathfrak a)\).