ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

As a prelude to the general theory of complex multiplication of abelian varieties, we discuss the arithmetic of elliptic curves with complex multiplication (CM). The goal of these notes is to provide a cohesive summary of the classical theory. The material presented here comes from a variety of sources, notably Gross [3], Lang [5], Serre [6], Shimura [7], and Silverman [9, Ch. II]; we make no claims to original-ity. We assume a familiarity with elliptic curves (our reference for basic facts is Silverman’s book [8]), algebraic geometry (on the level of Hartshorne’s book), and algebraic number theory (through class field theory). In order to avoid duplication of effort, we also assume the Main Theorem of CM (stated in §3), as it will be the subject of a later set of notes, [2, Thm. 7.3]