Non-abelian Fundamental Groups and Iwasawa Theory

Cover......Page 1
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES......Page 2
Title......Page 4
Copyright......Page 5
Contents......Page 6
Contributors......Page 7
Preface......Page 8
A. First examples......Page 12
B. Galois characterization of global fields......Page 13
A. Warm-up: birational anabelian conjectures......Page 16
B. Anabelian conjectures for curves......Page 21
C. The section conjectures......Page 44
Part III. Beyond the arithmetical action......Page 49
A. Small Galois groups and valuations......Page 50
B. Variation of fundamental groups in families of curves......Page 52
C. Pro- abelian-by-central birational anabelian geometry......Page 54
D. The Ihara/Oda–Matsumoto conjecture......Page 56
Some major open questions/problems......Page 59
Bibliography......Page 60
English translation of [31] (1989)......Page 67
Complementary notes......Page 73
References......Page 80
Introduction......Page 83
1 Generalities on arithmetic fundamental groups and sections......Page 85
2 Grothendieck anabelian section conjecture......Page 88
3 Good sections of arithmetic fundamental groups......Page 94
4 Cuspidalisation of sections of arithmetic fundamental groups......Page 100
5 Applications to the Grothendieck anabelian section conjecture......Page 108
6 On a weak form of the p-adic Grothendieck anabelian section conjecture......Page 112
References......Page 117
1 Introduction......Page 118
2 The set up......Page 120
3 The classical main conjecture......Page 121
3.1 The p-adic L-function......Page 122
3.3 The characteristic element......Page 123
4 Definition of K0 and K1......Page 124
5 The theory of determinants......Page 125
5.1 Definition of the category......Page 126
5.2 Extension to the derived category......Page 128
5.3 Change of rings......Page 129
5.5 Relation with characteristic elements......Page 130
6.1 Galois cohomology......Page 131
6.2 The Selmer complex......Page 132
6.3 The main conjecture......Page 133
6.4 The canonical Ore set......Page 135
6.1 Known results......Page 136
7 Generalisations......Page 138
References......Page 139
1 Introduction......Page 143
2 Statement of the conjecture......Page 144
3 Additional evidence for the MH(G)-conjecture......Page 151
4 Hida families over p-adic Lie extensions......Page 159
5 Analogue of the MH(G) conjecture for Hida families......Page 164
6 Vanishing of the R-torsion......Page 168
References......Page 171
1 The deficiency of abelian motives......Page 173
2 Motivic fundamental groups and Selmer varieties......Page 178
3 Diophantine finiteness......Page 184
4 An explicit formula and speculations......Page 190
References......Page 196
2 Semistable elliptic curves over Q are modular......Page 199
3 Why the semistability assumption?......Page 203
4 All elliptic curves over Q are modular......Page 206
5 Kisin’s modularity lifting theorems......Page 209
6 Generalisations to totally real fields......Page 212
7 Potential modularity pre-Kisin and the p–λ trick......Page 215
8 Potential modularity after Kisin......Page 218
9 Some final remarks......Page 219
References......Page 220
1 Introduction and notations......Page 223
2 Quick review of the GL2(Qp)-case......Page 226
3 Quick review of weakly admissible filtered ϕ-modules......Page 228
4 Some locally Qp -analytic representations of GL2 (F)......Page 231
5 Weak admissibility and GL2(F)-unitarity I......Page 234
6 Amice-V´ elu and Vishik revisited......Page 237
7 Weak admissibility and GL2(F)-unitarity II......Page 239
8 Local–global considerations......Page 243
9 The case where the Galois representation is reducible......Page 245
References......Page 248
Completed cohomology–a survey......Page 250
1 Definitions......Page 251
2 Non-commutative Iwasawa theory......Page 253
3 Poincare duality......Page 254
5 Congruence quotients of symmetric spaces......Page 255
6 Conjectures on codimensions......Page 257
7 Mod p analogues......Page 260
8 Heuristics related to the p-adic Langlands programme......Page 264
References......Page 267
1 Introduction......Page 269
2 Multi-Kummer characters......Page 272
3 Multi-Kummer duals......Page 275
4 Iterated integrals and their functional equations......Page 279
5 Case of polylogarithms......Page 293
6 Examples......Page 307
References......Page 318