**"Elliptic Curves and Explicit Class Field Theory"**

*Prof. Dr. Henri Darmon*

(McGill University, Canada)

Thursday, July 03. 2014, 17:15h

Mathematisches Institut, Hörsaal 2 Im Neuenheimer Feld 288

The values of the exponential function $e^{2\pi i z}$ and of the modular
function $j(z)$ (at rational and quadratic imaginary arguments, respectively)
lead to explicit generators for essentially all abelian extensions of the
rational numbers and of quadratic imaginary fields. The associated theories
of cyclotomic fields and of complex multiplication are quite rich and were
actively pursued in the 19th century. Kronecker's "Jugendtraum", raised again
by Hilbert as the twelfth in his celebrated list of open problems for the new
century, seeks to extend these theories to base fields other than the rationals
or quadratic imaginary fields. More than a hundred years later, Hilbert's 12th
problem is still largely open. This lecture shall survey the history of this
problem and describe some of the more recent attempts to make progress, based
on the study of elliptic curves and of their rational points.

In diesem Jahr berichtet der kanadische Mathematiker Henri Darmon, ein führender
Experte auf dem Gebiet der Zahlentheorie bzw. arithmetischen Geometrie, über
explizite Konstruktionen von Klassenkörpern (12. Problem von Hilbert) mittels
rationaler Punkte auf elliptischen Kurven.

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