RECENT DEVELOPMENTS IN THE THEORY OF COMPLEX MULTIPLICATION

Our story begins more than a century ago with Kronecker Jugendtraum and Hilbert 12th problem, where complex multiplication appears as a way to understand Galois extensions of number fields - a problem still at the heart of number theory. Our lecture will have a strong historical flavour; we shall attempt to survey the development of the theory of complex multiplication and the philosophy behind it. On this background, we will present some exciting recent results, some of which build in an essential way on Borcherds theory. Finally, we shall sketch some of the key challenges of the theory of complex multiplication today and future directions.