ABSTRACT
The endomorphisms of an elliptic curve E always include multiplication by arbitrary integers. When the endomorphism ring of E is strictly larger than Z, we say that E has complex multiplication. As we’ll see, elliptic curves over C with complex multiplication correspond to lattices with extra symmetry. Over finite fields, all elliptic curves have complex multiplication, and often the Frobenius provides one of the additional endomorphisms. In general, elliptic curves with complex multiplication form an interesting and important class of elliptic curves, partly because of their extra structure and partly because of their frequent occurrence.
