Value Distribution Theory Related to Number Theory

The subject of the book is Diophantine approximation and Nevanlinna theory. §This book proves not just some new results and directions but challenging open problems in Diophantine approximation and Nevanlinna theory. The authors newest research activities on these subjects over the past eight years are collected here. Some of the significant findings are the proof of Green-Griffiths conjecture by using meromorphic connections and Jacobian sections, generalized abc-conjecture, and more.The subject of the book is Diophantine approximation and Nevanlinna theory. Not only does the text provide new results and directions, it also challenges open problems and collects latest research activities on these subjects made by the authors over the past eight years. Some of the significant findings are the proof of the Green-Griffiths conjecture by using meromorphic connections and Jacobian sections, and a generalized abc-conjecture. The book also presents the state of the art in the studies of the analogues between Diophantine approximation (in number theory) and value distribution theory (in complex analysis), with a method based on Vojta s dictionary for the terms of these two fields. The approaches are relatively natural and more effective than existing methods. §The book is self-contained and appended with a comprehensive and up-to-date list of references. It is of interest to a broad audience of graduate students and researchers specialized in pure mathematics.