Einführung in die diophantische Approximation und Transzendenz

Beginn der Vorlesung: Freitag, 26.02.2010
Die erste Übungsstunde findet am Dienstag, 02.03.2010 statt.

Übungen: More details can be found here.

Synopsis

In the course we shall cover the basic techniques and results in transcendence theory. We shall begin with some elementary results on transcendence such as a construction of transcendental numbers which goes back to Liouville. Then we shall give a proof for the transcendence of e and pi.
After this we shall give the proof of Baker's qualitative theorem on linear forms in logarithms, which together with the criterion of Schneider and Lang is one of the most important results in number theory in the last century. We shall continue with proving the Schneider-Lang criterion and apply it to transcendence problems related to elliptic and abelian functions and varieties respectively. We shall also prove Lindemann's theorem on the algebraic independence of values
of the classical exponential function and towards the end of the course we give the proof of a qualitative version of Baker's theorem and apply it to problems in diophantine geometry.

Testatbedingungen

• Hand in at least 50% of the solutions to the problems on the exercise sheets (there will be an exercise sheet every second week approximately).
• Actively participate in the exercise sessions (e.g. by presenting at least once your solution to a problem or by giving a small talk). More information in the first exercise session on March 2, 2010.

Script

Additional handouts and lecture notes will be made available here.
Chapters 1-10 (Version of May 25, 2010): pdf

Literature

• Theodor Schneider, Einführung in die transzendenten Zahlen , Springer Verlag, 1957.
• Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975.
• Alan Baker, Gisbert Wüstholz, Logarithmic forms and diophantine geometry , New Mathematical Monographs, Cambridge University Press, 2007.