On Some Applications of Diophantine Approximations: A translation of C.L. Siegels Über einige Anwendungen diophantischer Approximationen, with a commentary by C. Fuchs and U. Zannier)

This book consists mainly of the translation, by C. Fuchs, of the 1929 landmark paper Über einige Anwendungen diophantischer Approximationen by C.L. Siegel. The paper contains proofs of most important results in transcendence theory and diophantine analysis, notably Siegels celebrated theorem on integral points on algebraic curves. Many modern versions of Siegels proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegels original ideas and proofs available for the first time in English. The volume also contains the original version of the paper (in German) and an article by the translator and U. Zannier, commenting on some aspects of the evolution of this field following Siegels paper. To end, it presents three modern proofs of Siegels theorem on integral points. See more