Preface ........................................................ xi
1 The Manin-Mumford Conjecture, an elliptic Curve, its
Torsion Points & their Galois Orbits ......................... 1
P. Habegger
1 Overview .................................................. 1
2 Elliptic Curves ........................................... 4
3 Galois Orbits of Torsion Points and Heights ............... 9
4 Application to the Manin-Mumford Conjecture .............. 27
Appendix. An Inequality of Elkies for the Local
Néron-Tate Height ........................................ 30
2 Rational points on definable sets ........................... 41
A.J. Wilkie
1 Introduction ............................................. 41
2 Some semi-algebraic geometry ............................. 42
3 O-minimal structures ([PS], [Dl]) ........................ 43
4 Some 1-dimensional o-minimal theory ...................... 46
5 Reparametrization (one variable case) .................... 47
6 Proof of 1.1 in the 1-dimensional case ................... 53
7 Some remarks on the proof of the general case of 1.1 ..... 58
8 Some higher-dimensional o-minimal theory ................. 59
9 Reparametrization (many variable case) ................... 61
3 Functional transcendence via o-minimality ................... 66
Jonathan Pila
1 Algebraic independence ................................... 66
2 Transcendental numbers ................................... 67
3 Schanuel's conjecture .................................... 69
4 Differential fields ...................................... 70
5 Ax-Schanuel .............................................. 71
6 "Ax-Lindemann" ........................................... 75
7 The modular function ..................................... 76
8 Modular Schanuel Conjecture .............................. 79
9 "Modular Ax-Schanuel" .................................... 80
10 "Modular Ax-Lindemann" ................................... 82
11 The general setting ...................................... 82
12 Exponential Ax-Lindemann via o-minimality ................ 84
13 Modular Ax-Lindemann via o-minimality .................... 87
14 SC and CIT ............................................... 91
15 Zilber-Pink .............................................. 94
16 Zilber-Pink and Ax-Schanuel .............................. 95
4 Introduction to abelian varieties and the Ax-Lindemann-
Weierstrass theorem ........................................ 100
Martin Orr
1 Introduction ............................................ 100
2 Abelian varieties ....................................... 102
3 Complex tori ............................................ 106
4 Riemann forms and polarisations ......................... 107
5 The moduli space of principally polarised abelian
varieties ............................................... 111
6 Complex multiplication .................................. 117
7 The Ax-Lindemann-Weierstrass theorem for abelian
varieties ............................................... 119
8 Relationship between semialgebraic and complex
algebraic sets .......................................... 121
9 Proof of the Ax-Lindemann-Weierstrass theorem for
abelian varieties ....................................... 123
5 The André-Oort conjecture via o-minimality ................. 129
Christopher Daw
1 Introduction ............................................ 129
2 Hermitian symmetric domains ............................. 131
3 Conjugacy classes ....................................... 133
4 The Deligne torus ....................................... 135
5 Hodge structures ........................................ 135
6 Abelian varieties ....................................... 137
7 The Siegel upper half-space ............................. 140
8 Families of Hodge structures ............................ 140
9 The algebraic group ..................................... 141
10 Shimuradata ............................................. 141
11 Congruence subgroups .................................... 143
12 Adeles .................................................. 144
13 Neatness ................................................ 144
14 Shimura varieties ....................................... 145
15 Complex structure ....................................... 145
16 Algebraic structure ..................................... 146
17 Special subvarieties .................................... 147
18 Special points .......................................... 147
19 Canonical model ......................................... 148
20 The André-Oort conjecture ............................... 150
21 Reductions .............................................. 150
22 Galois orbits ........................................... 151
23 Realisations ............................................ 152
24 Heights ................................................. 152
25 Definability ............................................ 154
26 Ax-Lindemann-Weierstrass ................................ 154
27 Pila-Wilkie ............................................. 155
28 Final reduction ......................................... 155
29 The Pila-Zannier strategy ............................... 155
6 Lectures on elimination theory for semialgebraic and
subanalytic sets ........................................... 159
A.J. Wilkie
1 Model Theoretic Generalities ............................ 160
2 The Real Field .......................................... 164
3 Preliminary Remarks on Rings and Modules ................ 166
4 Formal Power Series Rings ............................... 168
5 Adically Normed Rings ................................... 169
6 Formal Power Series in Many Variables ................... 172
7 Convergent Power Series ................................. 176
8 More on Adically Normed Rings and Modules ............... 181
9 The Denef-van den Dries Paper ........................... 185
7 Relative Manin-Mumford for abelian varieties ............... 193
D. Masser
8 Improving the bound in the Pila-Wilkie theorem
for curves ................................................. 204
G.O. Jones
9 Ax-Schanuel and o-minimality ............................... 216
Jacob Tsimerman
1 Interpreting Ax-Schanuel Geometrically .................. 216
2 An o-minimality proof of Ax-Schanuel .................... 218
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