The structure of the classifying ring of formal groups with complex multiplication

If $A$ is a commutative ring, there exists a classifying ring $L^A$ of formal groups with complex multiplication by $A$, i.e., "formal $A$-modules." In this paper, the basic properties of the functor that sends $A$ to $L^A$ are developed and studied. When $A$ is a Dedekind domain, the problem of computing $L^A$ was studied by M. Lazard, by V. Drinfeld, and by M. Hazewinkel, who showed that $L^A$ is a polynomial algebra whenever $A$ is a discrete valuation ring or a (global) number ring of class number $1$, Hazewinkel observed that $L^A$ is not necessarily polynomial for more general Dedekind domains $A$, but no computations of $L^A$ have ever appeared in any case when $L^A$ is not a polynomial algebra. In the present paper, the ring $L^A$ is computed, modulo torsion, for all Dedekind domains $A$ of characteristic zero, including many cases in which $L^A$ fails to be a polynomial algebra. Qualitative features (lifting and extensions) of the moduli theory of formal modules are then derived.