A panorama of number theory, or, The view from Baker's garden

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Contributors......Page 9
Introduction......Page 13
1 Introduction......Page 19
2 Hilbert’s seventh problem......Page 20
3 Elliptic theory......Page 21
4 Group varieties......Page 22
5 The quantitative theory......Page 23
6 The abc-conjecture......Page 25
Bibliography......Page 27
1 Historical introduction......Page 29
2 A successful strategy......Page 32
3 New developments......Page 35
4 The application to the abc-conjecture......Page 38
References......Page 40
1 Introduction......Page 44
2 New result......Page 48
3 Key estimate for the refinement......Page 49
References......Page 53
Introduction......Page 56
1 General effective finiteness theorems......Page 58
1.1 Thue equations and Thue–Mahler equations......Page 59
1.2 Elliptic, hyperelliptic and superelliptic equations......Page 62
1.3 Unit equations and S-unit equations......Page 64
1.4 Discriminant form and index form equations......Page 67
1.5 Decomposable form equations of general type......Page 71
2 Explicit determination of the solutions......Page 73
2.1 Thue equations......Page 76
2.2 Unit equations and S-unit equations......Page 77
2.3 Discriminant form and index form equations......Page 78
2.4 Decomposable form equations of general type......Page 81
2.5 Elliptic equations......Page 82
References......Page 85
1 Introduction......Page 91
2 Heights......Page 92
3 Baker’s method......Page 93
Baker’s theorem......Page 94
Functional units......Page 96
Coverings......Page 98
Preliminaries......Page 99
Effective Siegel theorem for modular curves......Page 100
Proof of Theorem 10......Page 102
References......Page 103
Abstract......Page 107
1 Some problems on hypergeometric functions......Page 108
2 Fuchsian triangle groups......Page 114
3 Solving Problem 4 solves Problem 1......Page 118
4 The André–Oort Conjecture for the Siegel moduli space......Page 119
References......Page 122
1 Triangle groups and dessins......Page 125
2 Integration on regular dessins......Page 128
3 Shimura families and the Jacobi locus......Page 131
References......Page 136
8 Maass Cusp Forms with Integer Coefficients......Page 139
Remarks......Page 142
Bibliography......Page 144
1 The ABC-Conjecture......Page 146
2 Applications of the ABC-Conjecture......Page 148
3 Elliptic curves over Q (Global Minimal Models)......Page 151
4 Conjectures which are equivalent to ABC......Page 152
6 Modular symbols......Page 160
References......Page 162
1 E-functions......Page 166
2 Mahler functions......Page 168
3 Theorems about functions with addition properties......Page 170
4 Modular forms......Page 172
5 Hypergeometric functions......Page 176
6 Values of theta functions......Page 179
References......Page 184
1 Definitions, notation and basic facts......Page 186
2 Integral ideal lattices......Page 189
Quadratic fields......Page 191
Cyclotomic fields of prime power conductor......Page 193
Examples over cyclotomic fields......Page 194
Knot theory......Page 195
Symmetric, skew-symmetric and orthogonal matrices with a given characteristic polynomial......Page 196
4 Real ideal lattices......Page 197
5 Arakelov invariants......Page 198
References......Page 201
1 Introduction......Page 203
2 Main results......Page 204
3 Examples......Page 206
4 Many integral points......Page 208
References......Page 210
1 Thue, Siegel and Mahler......Page 212
2 Baker......Page 214
3 Applications of Baker’s theory......Page 218
4 The Padé method......Page 221
5 An alternative effective method......Page 222
6 The future: the abc-conjecture......Page 224
References......Page 228
1 Introduction......Page 232
2 Proof of a weaker version of Corollary 1......Page 238
Absolute values and heights......Page 239
Points of small height......Page 241
The Subspace Theorem......Page 242
Proof of Theorem 7......Page 245
References......Page 246
1 Known results......Page 249
2 Geometry of projections......Page 252
3 Dimensions and expectation values......Page 258
4 Intersection numbers......Page 260
5 Nondegeneracy......Page 262
References......Page 264
16 Search Bounds for Diophantine Equations......Page 265
References......Page 275
1 Introduction......Page 278
2 Regular systems......Page 280
3 Ubiquity......Page 283
A general lower bound......Page 285
4 Khintchine-type theorems on manifolds......Page 287
5 Hausdorff dimension on manifolds......Page 292
Simultaneous Diophantine approximation on manifolds......Page 293
References......Page 294
1 Diophantine approximation on manifolds......Page 298
2 Values of quadratic forms and of products of linear forms at integral points......Page 307
3 A quantitative version of the Oppenheim conjecture......Page 312
4 Counting lattice points on homogeneous varieties......Page 319
5 Translates of submanifolds and unipotent flows on homogeneous spaces......Page 322
References......Page 325
Introduction and Baker’s Problem......Page 329
Some Extensions of Baker’s Problem......Page 331
Relations with Linnik’s Theorem......Page 332
Outline of the Proof of Theorem 5......Page 334
2 Vinogradov’s Bound......Page 338
References......Page 340
1 Cubes and higher powers......Page 343
2 Squares......Page 351
References......Page 352
21 On the Greatest Common Divisor of Two Univariate Polynomials, I......Page 355
References......Page 370
22 Heilbronn’s Exponential Sum and Transcendence Theory......Page 371
References......Page 374