The ends of finite-volume hyperbolic (n+1)-manifolds are cusps of the form B x R+ for some compact, flat n-manifold B.  In 2009, McReynolds built on work on Long and Reid to prove that every such n-manifold arises as the cusp cross section of some hyperbolic (n+1)-manifold using an arithmetic construction.  A natural further question to ask is under what conditions each cross-section can arise.  In this talk, we give an algebraic condition that describes exactly when a given flat manifold arises (as a cusp cross-section) in a commensurability class of cusped arithmetic hyperbolic manifolds.  Time permitting, we will also discuss some applications, including a potential new way to prove a hyperbolic manifold is non-arithmetic.  This is joint work with Duncan McCoy.