  Finite temperature boson and fermion field theories on ultrastatic
space-times with a d-sphere spatial section are discussed with one eye on the
questions of temperature inversion symmetry and modular invariance. For
conformally invariant theories it is shown that the total energy at any
temperature for any dimension, d, is given as a power series in the d=3 and d=5
energies, for scalars, and the d=1 and d=3 energies for spinors. Further, these
energies can be given in finite terms at specific temperatures associated with
singular moduli of elliptic function theory. Some examples are listed and
numbers given.
