An elliptic curve over C is said to have complex multiplication (CM) if it has more endomorphisms than multiplication by integers. In this talk, we will discuss the basic properties of CM curves and their relation with the class field theory of imaginary quadratic fields. Time permitting, I hope to sketch the l-adic proof of the integrality of j-invariant, a result related to the "conincidence" that e^{pi sqrt(163)} is very close to an integer. The idea is to use Neron-Ogg-Shafarevich criterion and local class field theory to show that any CM curve has good reduction after some base change, which is equivalent to having an integral j-invariant. This talk is aimed for learning, so I will try to give as many details and examples as possible.

Reference: Silverman AEC 1 and 2. Speaker(s): Yifeng Huang (UM)