Modular properties of massive scalar partition functions

We compute the exact thermal partition functions of a massive scalar field on flat spacetime backgrounds of the form Rd-q x Tq+1 and show that they possess an SL(q + 1, Z) symmetry. Non-trivial relations between equivalent expressions for the result are obtained by doing the computation using functional, canonical and worldline methods. For q = 1, the results exhibit modular symmetry and may be expressed in terms of massive Maass-Jacobi forms. In the complex case with chemical potential for U(1) charge turned on, the usual discussion of relativistic Bose-Einstein condensation is modified by the presence of the small dimensions.