Rational points and arithmetic of fundamental groups evidence for the section conjecture Jakob Stix

Format:

Book

Author/Creator:

Stix, Jakob

Series:

Lecture notes in mathematics (Springer-Verlag) 2054

Lecture notes in mathematics 1617-9692 2054

Language:

English

Subjects (All):

Geometry, Algebraic.

Number theory.

Non-Abelian groups.

Physical Description:

1 online resource

Place of Publication:

Berlin Springer ©2013

Language Note:

English

System Details:

text file

PDF

Summary:

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

Contents:

Part I Foundations of Sections

1 Continuous Non-abelian H1 with Profinite Coefficients.-2 The Fundamental Groupoid

3 Basic Geometric Operations in Terms of Sections

4 The Space of Sections as a Topological Space

5 Evaluation of Units

6 Cycle Classes in Anabelian Geometry

7 Injectivity in the Section Conjecture

Part II Basic Arithmetic of Sections

8 Reduction of Sections

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

Part III On the Passage from Local to Global

10 Local Obstructions at a p-adic Place

11 Brauer-Manin and Descent Obstructions

12 Fragments of Non-abelian Tate-Poitou Duality

Part IV Analogues of the Section Conjecture

13 On the Section Conjecture for Torsors

14 Nilpotent Sections

15 Sections over Finite Fields

16 On the Section Conjecture over Local Fields

17 Fields of Cohomological Dimension 1

18 Cuspidal Sections and Birational Analogues

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed October 24, 2012)

Other Format:

Printed edition:

ISBN:

364230673X

9783642306730

3642306748

9783642306747

OCLC:

814181066

Access Restriction:

Restricted for use by site license