Kubische Kurven : Von der Antike bis heute
Zahlentheorie ist ein Gebiet der Mathematik, das sich mit der Lösung von algebraischen Gleichungen mit ganzzahligen Koeffizienten beschäftigt. Sie befasst sich mit einer großen Anzahl von Problemen, die meist scheinbar einfach und elementar sind, aber manchmal sehr schwierig zu lösen.
The Birch and Swinnerton-Dyer conjecture (BSD), proposed in the 1960s, is one of the fundamental open questions in pure Mathematics. It establishes a connection between the algebraic properties of the rational points of an elliptic curve and the analytic features of its associated L-function. This article provides an informal and relatively elementary introduction to the circle of ideas surrounding BSD. It does so by first introducing the much older problem of congruent numbers, which dates back to antiquity. This problem, which is to date still open, can be reformulated as a question about the rational points of a certain class of elliptic curves. The validity of BSD would lead to a positive answer to the congruent number problem. In turn, this problem is used in the text as a convenient testing ground for the state of our knowledge on BSD. The article ends by recalling the fundamental role played by elliptic curves in the proof by Wiles and Taylor in the 1990’s of the famous Fermat’s Last Theorem. This proof relies on the deep and unexpected connection between elliptic curves and modular forms, which also implies the analytic continuation of the L-functions of elliptic curves.
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