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E-grāmata: Rational Points and Arithmetic of Fundamental Groups: Evidence for the Section Conjecture

  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2054
  • Izdošanas datums: 19-Oct-2012
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642306747
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Rational Points and Arithmetic of Fundamental Groups: Evidence for the Section ConjecturePalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2054
  • Izdošanas datums: 19-Oct-2012
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642306747
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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.

Recenzijas

From the book reviews:

The book under review, resulting from the authors dissertation is both a research monograph and a thorough presentation of the arithmetic and geometry of Grothendiecks 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)

Part I Foundations of Sections
1 Continuous Non-abelian H1 with Profinite Coefficients
3(10)
1.1 Torsors, Sections and Non-abelian H1
3(1)
1.2 Twisting, Differences and Comparison
4(3)
1.3 Long Exact Cohomology Sequence
7(2)
1.4 Non-abelian Hochschild-Serre Spectral Sequence
9(4)
2 The Fundamental Groupoid
13(12)
2.1 Fibre Functors and Path Spaces
13(1)
2.2 The Fundamental Groupoid
14(2)
2.3 The Fundamental Groupoid in the Relative Case
16(2)
2.4 Galois Invariant Base Points
18(2)
2.5 Abstract Sections
20(1)
2.6 Homotopy Fixed Points and the Section Conjecture
21(4)
3 Basic Geometric Operations in Terms of Sections
25(12)
3.1 Functoriality in the Space and Abelianization
25(1)
3.2 Base Change
26(1)
3.3 Centralisers and Galois Descent for Sections
27(1)
3.4 Fibres Above Sections
28(6)
3.5 Weil Restriction of Scalars and Sections
34(3)
4 The Space of Sections as a Topological Space
37(8)
4.1 Sections and Closed Subgroups
37(2)
4.2 Topology on the Space of Sections
39(1)
4.3 Limits of Sections
40(2)
4.4 The Decomposition Tower of a Section
42(3)
5 Evaluation of Units
45(8)
5.1 Kummer Theory with Finite Coefficients
45(2)
5.2 Kummer Theory with Profinite Coefficients
47(6)
6 Cycle Classes in Anabelian Geometry
53(16)
6.1 Various Definitions of Cycle Classes of a Section
53(4)
6.2 Algebraic K(π, 1) and Continuous Group Cohomology
57(1)
6.3 The Cycle Class of a Section Via Evaluation
58(2)
6.4 Anabelian Cycle Class for Subvarieties
60(2)
6.5 Parshin's π1-Approach to the Geometric Mordell Theorem
62(7)
Part II Basic Arithmetic of Sections
7 Injectivity in the Section Conjecture
69(12)
7.1 Injectivity via Arithmetic of Abelian Varieties
69(5)
7.2 Injectivity via Anabelian Assumptions
74(3)
7.3 Large Fundamental Group in the Sense of Kollar
77(4)
8 Reduction of Sections
81(14)
8.1 Specialisation
81(2)
8.2 The Ramification of a Section and Unramified Sections
83(3)
8.3 Bounds for the Cokernel of Specialisation
86(2)
8.4 Specialisation for Curves
88(4)
8.5 Specialisation for Sections Associated to Points
92(3)
9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers
95(12)
9.1 Compactness of the Space of Sections in the Arithmetic Case
95(2)
9.2 Weak Versus Strong: The Arithmetic Case
97(1)
9.3 Affine Versus Proper
98(1)
9.4 Centralisers of Sections in the Arithmetic Case
99(2)
9.5 Scalar Extensions and Geometric Covers
101(6)
Part III On the Passage from Local to Global
10 Local Obstructions at a p-adic Place
107(12)
10.1 Period and Index
107(4)
10.2 A Linear Form on the Lie-Algebra
111(6)
10.3 More Examples Exploiting the Relative Brauer Group
117(2)
11 Brauer-Manin and Descent Obstructions
119(28)
11.1 The Brauer-Manin Obstruction for Rational Points
119(1)
11.2 The Brauer-Manin Obstruction for Sections
120(5)
11.3 The Descent Obstruction for Rational Points
125(1)
11.4 Torsors Under Finite Etale Groups in Terms of Fundamental Groups
126(2)
11.5 Twisting
128(4)
11.6 The Descent Obstruction for Sections
132(5)
11.7 Consequences of Strong Approximation for Sections
137(6)
11.8 Generalizing the Descent Obstruction
143(1)
11.9 The Section Conjecture as an Only-One Conjecture
144(3)
12 Fragments of Non-abelian Tate-Poitou Duality
147(10)
12.1 Review of Non-abelian Galois Cohomology
147(1)
12.2 The Non-abelian Tate-Poitou Exact Sequence with Finite Coefficients
148(5)
12.3 The Non-abelian Tate-Poitou Exact Sequence with Profinite Coefficients
153(4)
Part IV Analogues of the Section Conjecture
13 On the Section Conjecture for Torsors
157(18)
13.1 The Fundamental Group of an Algebraic Group
157(2)
13.2 The Fundamental Group of a Torsor
159(3)
13.3 Examples
162(5)
13.4 The Weak Section Conjecture for Curves of Genus 0
167(4)
13.5 Zero Cycles and Abelian Sections
171(4)
14 Nilpotent Sections
175(22)
14.1 Primary Decomposition
175(1)
14.2 Obstructions from the Descending Central Series
176(4)
14.3 The Lie Algebra
180(3)
14.4 Finite Dimensional Subalgebras and Invariants
183(2)
14.5 Nilpotent Sections in the Arithmetic Case
185(1)
14.6 Pro-p Counter-Examples After Hoshi
186(6)
14.7 Variations on Pro-p Counter-Examples After Hoshi
192(5)
15 Sections over Finite Fields
197(10)
15.1 Abelian Varieties over Finite Fields
197(1)
15.2 Space Filling Curves in Their Jacobian
198(4)
15.3 Counting Sections
202(2)
15.4 Diophantine Sections After Tamagawa
204(3)
16 On the Section Conjecture over Local Fields
207(6)
16.1 The Real Section Conjecture
207(4)
16.2 The p-adic Section Conjecture
211(2)
17 Fields of Cohomological Dimension 1
213(6)
17.1 PAC Fields
213(2)
17.2 Infinite Algebraic Extensions of Finite Fields
215(2)
17.3 The Maximal Cyclotomic Extension of a Number Field
217(2)
18 Cuspidal Sections and Birational Analogues
219(14)
18.1 Construction of Tangential Sections
219(3)
18.2 The Characterisation of Cuspidal Sections for Curves
222(2)
18.3 Grothendieck's Letter
224(1)
18.4 The Birational Section Conjecture
225(4)
18.5 Birationally Liftable Sections
229(4)
List of Symbols 233(6)
References 239(8)
Index 247
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