11th October: Sebastian Monnet

Title: Complex Multiplication of Elliptic Curves

Abstract: This talk should be accessible to anyone who knows what a ring is and has seen number fields before. The endomorphisms of an elliptic curve naturally have the structure of a ring, and this ring always contains the integers. If this ring is actually bigger than the integers, then the elliptic curve is said to have "complex multiplication''. CM elliptic curves are often easier to work with than elliptic curves in general. For example, the Tate-Shafarevich conjecture has been proven for certain types of CM elliptic curves, whereas the general conjecture is still wide open. The goal of the talk is, assuming as few prerequisites as possible, to give some sense of what complex multiplication is and why you should care.