 The Ramanujan conjecture for automorphic forms on formula_70 over a number field would imply that for any congruence cover of an arithmetic three-manifold (derived from a quaternion algebra) the spectrum of the Laplace operator is contained in formula_71.Many of Thurston's conjectures (for example the virtually Haken conjecture), now all known to be true following the work of Ian Agol, were checked first for arithmetic manifolds by using specific methods.In some arithmetic cases the Virtual Haken conjecture is known by general means but it is not known if its solution can be arrived at by purely arithmetic means (for instance, by finding a congruence subgroup with positive first Betti number).