3. Emil Artin Lecture 2014

"Elliptic Curves and Explicit Class Field Theory"
Prof. Dr. Henri Darmon

Thursday, July 03. 2014, 17:15h
Mathematisches Institut, Hörsaal 2 Im Neuenheimer Feld 288

Abstract:

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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