401-4119-00L  Einführung in die diophantische Approximation und Transzendenz

SemesterFrühjahrssemester 2010
DozierendeG. Wüstholz
Periodizitäteinmalige Veranstaltung
LehrspracheDeutsch


KurzbeschreibungIn the course we shall cover the basic techniques and results in transcendence theory.
Lernziel
InhaltIn 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.
LiteraturThere are the "classics"

Th. Schneider, Einfuehrung in die transzendenten Zahlen, Springer Verlag (1957).
Alan Baker, Transcendental number theory, Cambridge Mathematical Libary, Cambridge University Press (1990).

The present state of art is documented in

Alan Baker, Gisbert Wuestholz, Logarithmic forms and diophantine geometry, New Mathematical Monographs, Cambridge University Press (2007).