We explore when a complete finite volume hyperbolic 3-manifold fibring over the circle is arithmetic. In the non-compact case, we show that there are only finitely many cyclic commensurability classes of arithmetic hyperbolic surface bundles with any given fibre type. We give a complete classification in the case of once-punctured torus bundles showing that there are precisely three cyclic commensurability classes. We give a partial result for compact manifolds.
Math. Annalen 302 (1995) 31-60.