Rational Points and Arithmetic of Fundamental Groups

1. Front Matter

   Pages i-xx

   PDF

2. Foundations of Sections

   1. Front Matter

      Pages 1-1

      PDF

   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

      PDF

   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

      PDF

   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

      PDF

   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

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.

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

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)

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

  Jakob Stix

Policies and ethics