Rational Points and Arithmetic of Fundamental Groups
Evidence for the Section Conjecture
Authors: Stix, Jakob
Free Preview- The first exposition of foundational material on the arithmetic of fundamental groups with respect to the Section Conjecture of anabelian Geometry: from the history of the subject to the state of the art of the conjecture.
- Numerous approaches to the Section Conjecture are discussed with open questions to stimulate future research.
- Assuming the basics, the more advanced chapters are self contained and can be read independently
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- About this book
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The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.
- Reviews
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From the book reviews:
“The book under review, resulting from the author’s dissertation … is both a research monograph and a thorough presentation of the arithmetic and geometry of Grothendieck’s section conjecture from the foundations to the current state of the art. … It will be useful not only to specialists, as it is accessible to anyone familiar with the basics of modern algebraic geometry and the theory of algebraic fundamental groups.” (Marco A. Garuti, Mathematical Reviews, May, 2014)
- Table of contents (18 chapters)
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Continuous Non-abelian H1 with Profinite Coefficients
Pages 3-11
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The Fundamental Groupoid
Pages 13-23
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Basic Geometric Operations in Terms of Sections
Pages 25-36
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The Space of Sections as a Topological Space
Pages 37-44
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Evaluation of Units
Pages 45-51
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Table of contents (18 chapters)
- Download Preface 1 PDF (58 KB)
- Download Sample pages 4 PDF (241.9 KB)
- Download Table of contents PDF (74.1 KB)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Rational Points and Arithmetic of Fundamental Groups
- Book Subtitle
- Evidence for the Section Conjecture
- Authors
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- Jakob Stix
- Series Title
- Lecture Notes in Mathematics
- Series Volume
- 2054
- Copyright
- 2013
- Publisher
- Springer-Verlag Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
- eBook ISBN
- 978-3-642-30674-7
- DOI
- 10.1007/978-3-642-30674-7
- Softcover ISBN
- 978-3-642-30673-0
- Series ISSN
- 0075-8434
- Edition Number
- 1
- Number of Pages
- XX, 249
- Topics
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