Marius Leonhardt

The main theorem(s) of complex multiplication and beyond

The theory of complex multiplication on elliptic curves has many
fascinating number-theoretic applications. For example, it can be used
to construct all abelian extensions of an imaginary quadratic number field.

At the heart of the theory lies the so-called main theorem of complex
multiplication. This talk will start by stating the main theorem and
then sketch the above-mentioned application.

Furthermore, it will explain how the main theorem can be used to
understand the Galois action on the moduli space of elliptic curves.
Time permitting, it will also discuss the higher-dimensional case of CM
abelian varieties, state various other versions of the main theorem and
mention a curious discovery by J. Nekovar.