I will sketch some classical analogies between the spectrum of Laplace type operators associated to Riemannian manifolds and the arithmetic of varieties defined over number fields.  I will speculate that the spectrum and arithmetic should mutually determine each other, provided we deal with 'canonical objects' on both sides.  Finally I will give some evidence supporting such wild speculations.  Towards the end, I will present some results on spectral analogues of the classical multiplicity one theorems for modular forms.