Reinier Broker (Microsoft Research)

Explicit complex multiplication theory

Abstract: The theory of complex multiplication links arithmetic geometry with algebraic number theory. This area of mathematics is rich with algorithms, and applications range from Hilbert's 12th problem to primality testing and cryptography.
In the first half of my talk I will explain the basic results from complex multiplication theory and indicate how to use them for various applications. During the second half, I will zoom in on the case of degree 4 CM-fields K. For such fields, the Igusa invariants of suitably chosen abelian surfaces generate an abelian extension of K. I will explain a new method to explicitly compute this extension, using the Galois action on the invariants coming from class field theory. Many examples will be given.