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

Cover......Page 1
Rational Points and Arithmetic of Fundamental Groups......Page 4
References......Page 8
Index......Page 10
Introduction......Page 14
1.1 Torsors, Sections and Non-abelian first cohomology......Page 22
1.2 Twisting, Differences and Comparison......Page 23
1.3 Long Exact Cohomology Sequence......Page 26
1.4 Non-abelian Hochschild–Serre Spectral Sequence......Page 28
2.1 Fibre Functors and Path Spaces......Page 31
2.2 The Fundamental Groupoid......Page 32
2.3 The Fundamental Groupoid in the Relative Case......Page 34
2.4 Galois Invariant Base Points......Page 36
2.5 Abstract Sections......Page 38
2.6 Homotopy Fixed Points and the Section Conjecture......Page 39
3.1 Functoriality in the Space and Abelianization......Page 42
3.2 Base Change......Page 43
3.3 Centralisers and Galois Descent for Sections......Page 44
3.4 Fibres Above Sections......Page 45
3.5 Weil Restriction of Scalars and Sections......Page 51
4.1 Sections and Closed Subgroups......Page 54
4.2 Topology on the Space of Sections......Page 56
4.3 Limits of Sections......Page 57
4.4 The Decomposition Tower of a Section......Page 59
5.1 Kummer Theory with Finite Coefficients......Page 62
5.2 Kummer Theory with Profinite Coefficients......Page 64
6.1 Various Definitions of Cycle Classes of a Section......Page 69
6.2 Algebraic K(pi,1) and Continuous Group Cohomology......Page 73
6.3 The Cycle Class of a Section Via Evaluation......Page 74
6.4 Anabelian Cycle Class for Subvarieties......Page 76
6.5 Parshin's pi1-Approach to the Geometric Mordell Theorem......Page 78
7.1 Injectivity via Arithmetic of Abelian Varieties......Page 83
7.2 Injectivity via Anabelian Assumptions......Page 88
7.3 Large Fundamental Group in the Sense of Kollár......Page 91
8.1 Specialisation......Page 94
8.2 The Ramification of a Section and Unramified Sections......Page 96
8.3 Bounds for the Cokernel of Specialisation......Page 99
8.4 Specialisation for Curves......Page 101
8.5 Specialisation for Sections Associated to Points......Page 105
9.1 Compactness of the Space of Sections in the Arithmetic Case......Page 107
9.2 Weak Versus Strong: The Arithmetic Case......Page 109
9.3 Affine Versus Proper......Page 110
9.4 Centralisers of Sections in the Arithmetic Case......Page 111
9.5 Scalar Extensions and Geometric Covers......Page 113
10.1 Period and Index......Page 116
10.2 A Linear Form on the Lie-Algebra......Page 120
10.3 More Examples Exploiting the Relative Brauer Group......Page 126
11.1 The Brauer–Manin Obstruction for Rational Points......Page 127
11.2 The Brauer–Manin Obstruction for Sections......Page 128
11.3 The Descent Obstruction for Rational Points......Page 133
11.4 Torsors Under Finite Étale Groups in Terms of Fundamental Groups......Page 134
11.5 Twisting......Page 136
11.6 The Descent Obstruction for Sections......Page 140
11.7 Consequences of Strong Approximation for Sections......Page 145
11.8 Generalizing the Descent Obstruction......Page 151
11.9 The Section Conjecture as an Only–One Conjecture......Page 152
12.1 Review of Non-abelian Galois Cohomology......Page 155
12.2 The Non-abelian Tate–Poitou Exact Sequence with Finite Coefficients......Page 156
12.3 The Non-abelian Tate–Poitou Exact Sequence with Profinite Coefficients......Page 161
13.1 The Fundamental Group of an Algebraic Group......Page 163
13.2 The Fundamental Group of a Torsor......Page 165
13.3 Examples......Page 168
13.4 The Weak Section Conjecture for Curves of Genus zero......Page 173
13.5 Zero Cycles and Abelian Sections......Page 177
14.1 Primary Decomposition......Page 181
14.2 Obstructions from the Descending Central Series......Page 182
14.3 The Lie Algebra......Page 186
14.4 Finite Dimensional Subalgebras and Invariants......Page 189
14.5 Nilpotent Sections in the Arithmetic Case......Page 191
14.6 Pro-p Counter-Examples After Hoshi......Page 192
14.7 Variations on Pro-p Counter-Examples After Hoshi......Page 198
15.1 Abelian Varieties over Finite Fields......Page 203
15.2 Space Filling Curves in Their Jacobian......Page 204
15.3 Counting Sections......Page 208
15.4 Diophantine Sections After Tamagawa......Page 210
16.1 The Real Section Conjecture......Page 212
16.2 The p-adic Section Conjecture......Page 216
17.1 PAC Fields......Page 218
17.2 Infinite Algebraic Extensions of Finite Fields......Page 220
17.3 The Maximal Cyclotomic Extension of a Number Field......Page 222
18.1 Construction of Tangential Sections......Page 224
18.2 The Characterisation of Cuspidal Sections for Curves......Page 227
18.3 Grothendieck's Letter......Page 229
18.4 The Birational Section Conjecture......Page 230
18.5 Birationally Liftable Sections......Page 234