Perspectives of Diophantine approximation and transcendence
type: Item
instance of: scholarly article
title: Perspectives of Diophantine approximation and transcendence
zbMATH Open document ID: 1049.11034
DOI: 10.1023/A:1026271918122
published in: The Ramanujan Journal
publication date: 2004-01-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J89
Mathematics Subject Classification ID: 11J91
zbMATH DE Number: 2022038
zbMATH Keywords: Padé approximation
zbMATH Keywords: Thue-Siegel principle
zbMATH Keywords: diophantine approximation of algebraic numbers
author: Paula Tretkoff
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
full work available at URL: a:1026271918122
OpenAlex ID: W4250883644
description: scientific article; zbMATH DE number 2022038