The Cohomology of Hilbert Modular Surfaces

Abstract

All interesting cohomology of Hilbert modular surfaces lies in the middle dimension, i.e. in H ². The main part of this is the cuspidal cohomology; it carries a Hodge structure and its spaces of forms of type (p, q), p + q = 2, are described by spaces of Hilbert modular forms of weight 2. The cohomology group ℍ²(X _Γ ) is the cohomology generated by the cuspidal cohomology ℍ ² _sp (X _Γ ) and the classes of the Chern forms w _i associated to the automorphy factors (cz _i + d) ² (i = 1, 2). It can also be described as intersection homology of X _Γ .