  We study N = 2* theories with gauge group U(N) and use equivariant
localization to calculate the quantum expectation values of the simplest chiral
ring elements. These are expressed as an expansion in the mass of the adjoint
hypermultiplet, with coefficients given by quasi-modular forms of the S-duality
group. Under the action of this group, we construct combinations of chiral ring
elements that transform as modular forms of definite weight. As an independent
check, we confirm these results by comparing the spectral curves of the
associated Hitchin system and the elliptic Calogero-Moser system. We also
propose an exact and compact expression for the 1-instanton contribution to the
expectation value of the chiral ring elements.
