Philipp Habegger (Universität Basel) - Mardi 16 mai 2017 (14 heures)
Complex Multiplication or is $e^{\pi \sqrt{163}}$ an Integer?
Roots of unity are algebraic values of the exponential
function at rational multiples of $i\pi$. 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 Pila and more recently with Bilu-Kühne.
(Lire la suite...)
