Perspectives of Diophantine approximation and transcendence
In this lecture the author gives an excellent survey of selected aspects of Diophantine approximation and transcendental numbers. The first one is related with the open problem of proving an effective version of Siegel's Theorem on the finiteness of integral points on curves of genus \(\geq 2\). Her description includes an introduction to Bombieri's new approach for proving effective irrationality measures for algebraic numbers better than Liouville's one. The \(abc\) conjecture of Œsterlé and Masser is also discussed, in connection with lower bounds for linear forms in logarithms of algebraic numbers (conjectures of Baker and Philippon). The last section is a valuable discussion of various questions connected with Schneider's problem asking to prove transcendence results on special values of modular functions without using Abelian varieties. This section includes the solution by Barré-Sirieix, Diaz, Gramain and Philibert of a conjecture of Mahler and Manin as well as the algebraic independence result on values of Eisenstein series due to Nesterenko. The main topic of this section is the André-Oort conjecture with references to the recent results by Y.~André, P. B. Cohen, H.~Shiga, J.~Wolfart, G.~Wüstholz, B. Edixhoven and A. Yafaev.
diophantine approximation of algebraic numbers
scientific article; zbMATH DE number 2022038