Let K be a number field with ring of integers \(R = \mathcal {O}_K\) . We show that if R is not a principal ideal domain, then the symplectic group \(\operatorname {Sp}_{2n}(R)\) has non-trivial rational cohomology in its virtual cohomological dimension. This demonstrates a sharp contrast to the situation where R is Euclidean. To prove our result, we study the symplectic Steinberg module, i.e. the top-dimensional homology group of the spherical building associated to \(\operatorname {Sp}_{2n}(K)\) . We show that this module is not generated by integral apartment classes. Both of these results follow from a vanishing theorem for homology with Steinberg coefficients.
