  Machinery is developed for computing the classifying ring $L^A$ of
one-dimensional formal groups with complex multiplication by $A$, for a given
commutative ring $A$. The machinery is then applied to compute $L^A$ for
various number rings and cyclic group rings $A$. The ring $L^A$ has been
computed, for certain choices of $A$, by M. Lazard, V. Drinfeld, and M.
Hazewinkel, but in those cases $L^A$ is always isomorphic to a polynomial
algebra. In the present paper, $L^A$ is computed in many cases in which it
fails to be a polynomial algebra, leading to a qualitatively different moduli
theory and a different presentation for the moduli stack of formal $A$-modules.
