The theory of elliptic curves with large endomorphism rings. For questions on multiplication of complex numbers, use (complex-numbers) instead.

According to Wikipedia:

Complex multiplication is the theory of elliptic curves $E$ that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties $A$ having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of $A$ is a direct sum of one-dimensional modules). Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

It is useful in the study of class field theory.