Cusps of Hilbert modular varieties.

D. B. McReynolds

Let M denote a virtual n-torus bundle over an m-torus. In this article we give a necessary and sufficient condition for M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. One application of this classification theorem answers an implicit question of Hirzebruch on the possible isomorphism types of the fundamental group of a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every orientable solvable 3-manifold or a 2-fold cover of it is diffeomorphic to a cusp cross-section of a Hilbert modular surface. We conclude this article by proving
that certain solvable 3-manifolds cannot arise as the cusp cross-section of a 1-cusped nonsingular Hilbert modular surface.