401-4119-00L Einführung in die diophantische Approximation und Transzendenz
Semester | Frühjahrssemester 2010 |
Dozierende | G. Wüstholz |
Periodizität | einmalige Veranstaltung |
Lehrsprache | Deutsch |
Kurzbeschreibung | In the course we shall cover the basic techniques and results in transcendence theory. |
Lernziel | |
Inhalt | 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. |
Literatur | There 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). |