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Rational Points and Arithmetic of Fundamental Groups

Evidence for the Section Conjecture

  • Book
  • © 2013

Overview

Authors:
  1. Jakob Stix

    1. MATCH - Mathematics Center Heidelberg, Department of Mathematics, University of Heidelberg, Heidelberg, Germany

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  • 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
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2054)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xx
    Download chapter PDF

  2. Foundations of Sections

    1. Front Matter

      Pages 1-1
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    2. Continuous Non-abelian H1 with Profinite Coefficients

      • Jakob Stix
      Pages 3-11

    3. The Fundamental Groupoid

      • Jakob Stix
      Pages 13-23

    4. Basic Geometric Operations in Terms of Sections

      • Jakob Stix
      Pages 25-36

    5. The Space of Sections as a Topological Space

      • Jakob Stix
      Pages 37-44

    6. Evaluation of Units

      • Jakob Stix
      Pages 45-51

    7. Cycle Classes in Anabelian Geometry

      • Jakob Stix
      Pages 53-66

  3. Basic Arithmetic of Sections

    1. Front Matter

      Pages 67-67
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    2. Injectivity in the Section Conjecture

      • Jakob Stix
      Pages 69-79

    3. Reduction of Sections

      • Jakob Stix
      Pages 81-93

    4. The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers

      • Jakob Stix
      Pages 95-103

  4. On the Passage from Local to Global

    1. Front Matter

      Pages 105-105
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    2. Local Obstructions at a p-adic Place

      • Jakob Stix
      Pages 107-117

    3. Brauer–Manin and Descent Obstructions

      • Jakob Stix
      Pages 119-146

    4. Fragments of Non-abelian Tate–Poitou Duality

      • Jakob Stix
      Pages 147-154

  5. Analogues of the Section Conjecture

    1. Front Matter

      Pages 155-155
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    2. On the Section Conjecture for Torsors

      • Jakob Stix
      Pages 157-174

    3. Nilpotent Sections

      • Jakob Stix
      Pages 175-196

    4. Sections over Finite Fields

      • Jakob Stix
      Pages 197-205
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Keywords

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)

Authors and Affiliations

  • MATCH - Mathematics Center Heidelberg, Department of Mathematics, University of Heidelberg, Heidelberg, Germany

    Jakob Stix

Bibliographic Information

  • Book Title: Rational Points and Arithmetic of Fundamental Groups

  • Book Subtitle: Evidence for the Section Conjecture

  • Authors: Jakob Stix

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-30674-7

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2013

  • Softcover ISBN: 978-3-642-30673-0Published: 19 October 2012

  • eBook ISBN: 978-3-642-30674-7Published: 19 October 2012

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XX, 249

  • Topics: Algebraic Geometry, Number Theory

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