Applications of Diophantine Approximation to Integral Points and Transcendence
Title | Applications of Diophantine Approximation to Integral Points and Transcendence PDF eBook |
Author | Pietro Corvaja |
Publisher | Cambridge University Press |
Pages | 209 |
Release | 2018-05-03 |
Genre | Mathematics |
ISBN | 1108424945 |
Introduction to Diophantine approximation and equations focusing on Schmidt's subspace theorem, with applications to transcendence.
On Some Applications of Diophantine Approximations
Title | On Some Applications of Diophantine Approximations PDF eBook |
Author | Umberto Zannier |
Publisher | Springer |
Pages | 169 |
Release | 2015-02-13 |
Genre | Mathematics |
ISBN | 8876425209 |
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 Siegel’s celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel’s proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel’s 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 Siegel’s paper. To end, it presents three modern proofs of Siegel’s theorem on integral points.
New Advances in Transcendence Theory
Title | New Advances in Transcendence Theory PDF eBook |
Author | Alan Baker |
Publisher | Cambridge University Press |
Pages | 456 |
Release | 1988-10-13 |
Genre | Mathematics |
ISBN | 9780521335454 |
This is an account of the proceedings of a very successful symposium of Transcendental Number Theory held in Durham in 1986. Most of the leading international specialists were present and the lectures reflected the great advances that have taken place in this area. The papers cover all the main branches of the subject, and include not only definitive research but valuable survey articles.
Diophantine Approximation and Its Applications
Title | Diophantine Approximation and Its Applications PDF eBook |
Author | Charles F. Osgood |
Publisher | |
Pages | 355 |
Release | 1973 |
Genre | |
ISBN |
Transcendence and Linear Relations of 1-Periods
Title | Transcendence and Linear Relations of 1-Periods PDF eBook |
Author | Annette Huber |
Publisher | Cambridge University Press |
Pages | 266 |
Release | 2022-05-26 |
Genre | Mathematics |
ISBN | 1009022717 |
This exploration of the relation between periods and transcendental numbers brings Baker's theory of linear forms in logarithms into its most general framework, the theory of 1-motives. Written by leading experts in the field, it contains original results and finalises the theory of linear relations of 1-periods, answering long-standing questions in transcendence theory. It provides a complete exposition of the new theory for researchers, but also serves as an introduction to transcendence for graduate students and newcomers. It begins with foundational material, including a review of the theory of commutative algebraic groups and the analytic subgroup theorem as well as the basics of singular homology and de Rham cohomology. Part II addresses periods of 1-motives, linking back to classical examples like the transcendence of π, before the authors turn to periods of algebraic varieties in Part III. Finally, Part IV aims at a dimension formula for the space of periods of a 1-motive in terms of its data.
On Some Applications of Diophantine Approximations
Title | On Some Applications of Diophantine Approximations PDF eBook |
Author | Carl Ludwig Siegel |
Publisher | |
Pages | 161 |
Release | 2014 |
Genre | Diophantine analysis |
ISBN |
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 Siegel's celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel's proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel's 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 Siegel's paper. To end, it presents three modern proofs of Siegel's theorem on integral points.
Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces
Title | Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces PDF eBook |
Author | Marc-Hubert Nicole |
Publisher | Springer Nature |
Pages | 247 |
Release | 2020-10-31 |
Genre | Mathematics |
ISBN | 3030498646 |
This textbook introduces exciting new developments and cutting-edge results on the theme of hyperbolicity. Written by leading experts in their respective fields, the chapters stem from mini-courses given alongside three workshops that took place in Montréal between 2018 and 2019. Each chapter is self-contained, including an overview of preliminaries for each respective topic. This approach captures the spirit of the original lectures, which prepared graduate students and those new to the field for the technical talks in the program. The four chapters turn the spotlight on the following pivotal themes: The basic notions of o-minimal geometry, which build to the proof of the Ax–Schanuel conjecture for variations of Hodge structures; A broad introduction to the theory of orbifold pairs and Campana's conjectures, with a special emphasis on the arithmetic perspective; A systematic presentation and comparison between different notions of hyperbolicity, as an introduction to the Lang–Vojta conjectures in the projective case; An exploration of hyperbolicity and the Lang–Vojta conjectures in the general case of quasi-projective varieties. Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces is an ideal resource for graduate students and researchers in number theory, complex algebraic geometry, and arithmetic geometry. A basic course in algebraic geometry is assumed, along with some familiarity with the vocabulary of algebraic number theory.