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© 2013

Rational Points and Arithmetic of Fundamental Groups

Evidence for the Section Conjecture

Book

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

Table of contents

  1. Front Matter
    Pages i-xx
  2. Foundations of Sections

    1. Front Matter
      Pages 1-1
    2. Jakob Stix
      Pages 13-23
    3. Jakob Stix
      Pages 45-51
    4. Jakob Stix
      Pages 53-66
  3. Basic Arithmetic of Sections

    1. Front Matter
      Pages 67-67
    2. Jakob Stix
      Pages 69-79
    3. Jakob Stix
      Pages 81-93
  4. On the Passage from Local to Global

    1. Front Matter
      Pages 105-105
    2. Jakob Stix
      Pages 107-117
    3. Jakob Stix
      Pages 119-146
  5. Analogues of the Section Conjecture

    1. Front Matter
      Pages 155-155
    2. Jakob Stix
      Pages 157-174
    3. Jakob Stix
      Pages 175-196
    4. Jakob Stix
      Pages 197-205

About this book

Introduction

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.

Keywords

14H30,14G05,14H25,11G20,14G32,14F35. Anabelian Geometry Etale Fundamental Group Rational Points Section Conjecture

Authors and affiliations

  1. 1.MATCH - Mathematics Center Heidelberg, Department of MathematicsUniversity of HeidelbergHeidelbergGermany

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
  • Series Abbreviated Title Lect.Notes Mathematics
  • DOI https://doi.org/10.1007/978-3-642-30674-7
  • Copyright Information Springer-Verlag Berlin Heidelberg 2013
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-642-30673-0
  • eBook ISBN 978-3-642-30674-7
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages XX, 249
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Algebraic Geometry
    Number Theory
  • Buy this book on publisher's site
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Finance, Business & Banking

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)