Elliptic functions and temperature inversion symmetry on spheres

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.