Lecture Notes in Mathematics

Rational Points and Arithmetic of Fundamental Groups

Evidence for the Section Conjecture

Authors: Stix, Jakob

  • 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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Softcover 46,75 €
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About this book

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

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)

  • Continuous Non-abelian H1 with Profinite Coefficients

    Stix, Jakob

    Pages 3-11

    Preview Buy Chapter 30,19 €
  • The Fundamental Groupoid

    Stix, Jakob

    Pages 13-23

    Preview Buy Chapter 30,19 €
  • Basic Geometric Operations in Terms of Sections

    Stix, Jakob

    Pages 25-36

    Preview Buy Chapter 30,19 €
  • The Space of Sections as a Topological Space

    Stix, Jakob

    Pages 37-44

    Preview Buy Chapter 30,19 €
  • Evaluation of Units

    Stix, Jakob

    Pages 45-51

    Preview Buy Chapter 30,19 €

Buy this book

eBook 35,69 €
price for Spain (gross)
  • ISBN 978-3-642-30674-7
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover 46,75 €
price for Spain (gross)
  • ISBN 978-3-642-30673-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Rent the eBook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Rational Points and Arithmetic of Fundamental Groups
Book Subtitle
Evidence for the Section Conjecture
Authors
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