Hilbert's tenth problem: Diophantine classes and extensions to global fields

Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Copyright......Page 6
Title......Page 7
Dedication......Page 9
Contents......Page 11
Acknowledgements......Page 15
1.1 In the beginning......Page 17
1.2 Diophantine definitions and Diophantine sets......Page 20
2.1 Diophantine generation......Page 25
2.2 Diophantine generation of integral closure and Dioph-regularity......Page 39
2.3 Big picture: Diophantine family of a ring......Page 41
3.1 Weak presentations......Page 45
3.2 Some properties of weak presentations......Page 46
3.3 How many Diophantine classes are there?......Page 52
3.4 Diophantine generation and Hilbert’s Tenth Problem......Page 53
4 Integrality at finitely many primes and divisibility of order at infinitely many primes......Page 60
4.1 The main ideas......Page 61
4.2 Integrality at finitely many primes in number fields......Page 62
4.3 Integrality at finitely many primes over function fields......Page 69
4.4 Divisibility of order at infinitely many primes over number fields......Page 73
4.5 Divisibility of order at infinitely many primes over function fields......Page 80
5.1 Real embeddings......Page 82
5.2 Using divisibility in the rings of algebraic integers......Page 83
5.3 Using divisibility in bigger rings......Page 86
6.1 What are the units of the rings of W-integers?......Page 91
6.2 Norm equations of units......Page 93
6.3 The Pell equation......Page 96
6.4 Non-integral solutions of some unit norm equations......Page 107
7.1 Vertical methods of Denef and Lipshitz......Page 112
7.2 Integers of totally real number fields and fields with exactly one pair of non-real embeddings......Page 114
7.3 Integers of extensions of degree 2 of totally real number fields......Page 116
7.4 The main results for the rings of W-integers and an overview of the proof......Page 119
7.5 The main vertical definability results for rings of W-integers in totally real number fields......Page 121
7.6 Consequences for vertical definability over totally real fields......Page 127
7.7 Horizontal definability for rings of W-integers of totally real number fields and Diophantine undecidability for these rings......Page 130
7.8 Vertical definability results for rings of W-integers of the totally complex extensions of degree 2 of totally real number fields......Page 131
7.9 Some consequences......Page 136
7.10 Big picture for number fields revisited......Page 140
7.11 Further results......Page 143
8.1 Defining multiplication through localized divisibility......Page 145
8.2 pth power equations over function fields I: Overview and preliminary results......Page 150
8.3 pth power equations over function fields II: pth powers of a special element......Page 161
8.4 pth power equations over function fields III: pth powers of arbitrary functions......Page 167
8.5 Diophantine model of Z over function fields over finite fields of constants......Page 175
9.1 Height bounds......Page 178
9.2 Using pth powers to bound the height......Page 180
10.1 The weak vertical method revisited......Page 182
10.2 The weak vertical method applied to non-constant cyclic extensions......Page 183
10.3 The weak vertical method applied to constant field extensions......Page 187
10.4 Vertical definability for large subrings of global function fields......Page 188
10.5 Integrality at infinitely many primes over global function fields......Page 191
10.6 The big picture for function fields revisited......Page 192
10.6.1. Notation and assumptions......Page 193
11.1 The two conjectures......Page 196
11.2 A ring version of Mazur’s first conjecture......Page 197
11.3 First counterexamples......Page 199
11.4 Consequences for Diophantine models......Page 202
12.1 A statement of the main theorem and an overview of the proof......Page 205
12.2 Properties of elliptic curves I: Factors of denominators of points......Page 207
12.3 Properties of elliptic curves II: The density of the set of “largest” primes......Page 209
12.4 Properties of elliptic curves III: Finite sets looking big......Page 214
12.6 Construction of the sets T1(P) and T2(P) and their properties......Page 216
12.7 Proof of Poonen’s theorem......Page 222
13.1 Function fields of positive characteristic and of higher transcendence degree or over infinite fields of constants......Page 225
13.2 Algebraic extensions of global fields of infinite degree......Page 227
13.3 Function fields of characteristic 0......Page 228
A.1 Computable (recursive) functions......Page 231
A.2 Recursively enumerable sets......Page 238
A.3 Turing and partial degrees......Page 239
A.4 Degrees of sets of indices, primes, and products......Page 240
A.5 Recursive algebra......Page 241
A.6 Recursive presentation of Q......Page 242
A.7 Recursive presentation of other fields......Page 247
A.8 Representing sets of primes and rings of S-integers in number fields......Page 252
B.1 Global fields, valuations, and rings of W-integers......Page 257
B.2 Existence through approximation theorems......Page 267
B.3 Linearly disjoint fields......Page 269
B.4 Divisors, prime and composite, under extensions......Page 273
B.5 Density of prime sets......Page 293
B.6 Elliptic curves......Page 305
B.7 Coordinate polynomials......Page 312
B.8 Basic facts about local fields......Page 316
B.9 Derivations......Page 317
B.10 Some calculations......Page 319
References......Page 326
Index......Page 333