Elliptic Curves have motivated a large part of the mathematics of the last century: in particular, they have allowed for the famous proof of Fermat's Last Theorem. They still are an area of active research, in pure mathematics (for example, motivated by the Birch and Swinnerton-Dyer conjecture, one of the seven Clay Math problems) but also in applied mathematics in relation to many applications in cryptography.

Over the complex numbers, elliptic curves have a particularly simply description: an elliptic curve is just a complex torus. But this conceals a lot of their inner structure! Trying to classify them up to isomorphism leads into many interesting problems, such as the idea of moduli spaces, the mysterious j-invariant and doubly periodic functions on the complex plane.

These elliptic curves come with a group structure (either coming from their description as a torus or by considering a chord-tangent group law, the two being related by the Weierstrass function), and we can thus consider their endomorphisms (the group homomorphisms from the elliptic curve to itself). Every elliptic curve over the complex numbers comes with a multiplication by endomorphism for every integer , but for most of them, that's the end of the story. Special elliptic curves, with more endomorphisms, are said to have complex multiplication.

The study of elliptic curves with complex multiplication relates perhaps surprisingly with class field theory and the study of quadratic imaginary fields (fields of the form for positive integers ); there is a direct relation between isomorphism classes of elliptic curves with given endomorphisms and corresponding quadratic imaginary fields.

Join us in MS.04 at 7:30 to learn about all this and more! Don't miss it - this is probably the last Discussion Group for a while due to exams starting for many of us. We will of course translate ourselves to the pub afterwards.