"Roots of unity are algebraic values of the exponential
function at rational arguments shifted by $\pi i$. Kronecker's Jugendtraum was
to find analytic functions that mimic this behavior for algebraic numbers of higher degree.
The theory of complex multiplication of elliptic curves provides a rich
trove of examples of such functions with many surprising symmetries. It
originated in the 19th century in work of Kronecker and Weber and
underwent a remarkable development in the 20th century by Hilbert,
Shimura, Deligne and many others.

In this talk I will provide a glimpse into some classical aspects of complex multiplication
from a diophantine point of view. Then I will discuss recent questions
connected to problems in diophantine geometry, some of them are joint
work with Jonathan Pila.