Cover Page
The handle
http://hdl.handle.net/1887/82075
holds various files of this Leiden University
dissertation.
The unit residue group
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van Rector Magnificus prof. mr. C. J. J. M. Stolker,
volgens besluit van het College voor Promoties
te verdedigen op woensdag 18 december 2019
klokke 12:30 uur
door
Gabriele Dalla Torre
,
geboren te Trento, Itali¨
e,
Promotor
prof. dr. H. W. Lenstra
Promotiecommissie
prof. dr. P. Stevenhagen (voorzitter, Universiteit Leiden) prof. dr. R. M. van Luijk (secretaris, Universiteit Leiden) prof. dr. Roberto Dvornicich (Universit`a di Pisa)
The unit residue group
Gabriele Dalla Torre, Leiden, 2019 gabrieledallatorre@gmail.com
Contents
1 Introduction 1
1.1 Quadratic reciprocity . . . 2
1.2 The Jacobi symbol . . . 3
1.3 Orthogonal primes . . . 5
1.4 The norm-residue symbol . . . 7
1.5 Orthogonality . . . 10
1.6 Global formulation . . . 12
1.7 Overview . . . 15
1.8 Skew abelian groups . . . 16
1.9 The local norm-residue symbol . . . 17
1.10 The global norm-residue symbol . . . 18
1.11 The unit residue group . . . 19
1.12 Quadratic number fields . . . 20
1.13 Two-ranks of ideal class groups . . . 21
1.14 Biquadratic number fields . . . 22
1.15 Cyclic number fields . . . 22
1.16 A large norm group . . . 23
1.17 Quadratic characters . . . 24
Contents
3 The local norm-residue symbol 45
3.1 Topological algebra . . . 45
3.2 Local fields . . . 47
3.3 Abelian Kummer theory . . . 51
3.4 Local class field theory . . . 53
3.5 The norm-residue symbol . . . 57
3.6 A new elementary characterization . . . 61
3.7 Archimedean local fields . . . 63
3.8 Non-Archimedean local fields . . . 64
3.9 Functorial properties . . . 72
3.10 The field of two-adic rationals and its unramified extensions . . 72
4 The global norm-residue symbol 77 4.1 Global fields . . . 77
4.2 Adeles and ideles . . . 79
4.3 Locally compact abelian groups . . . 82
4.4 Self-duality . . . 85
4.5 Function fields . . . 88
4.6 The Tate pairing . . . 93
4.7 The Arakelov class group . . . 95
4.8 Number fields . . . 98
5 The unit residue group 103 5.1 Definitions and general results . . . 104
5.2 The virtual group . . . 111
5.3 The field of rational numbers . . . 115
6 Quadratic number fields and a biquadratic example 117 6.1 Introduction . . . 117
6.2 Discriminants of quadratic number fields . . . 119
6.3 Unramified extensions . . . 119
6.4 The two-adic component of the unit residue group . . . 122
6.5 Imaginary quadratic number fields . . . 125
6.6 Real quadratic number fields . . . 129
6.7 The number field Q(i,√30) . . . 134
7 Two-ranks of ideal class groups 137 7.1 Results . . . 137
7.2 Armitage–Fr¨ohlich’s theorem . . . 139
7.3 Proof of the main theorem . . . 140
Contents
8 Three-ranks of ideal class groups of quadratic number fields 143
8.1 Introduction . . . 143
8.2 Two results on modules . . . 146
8.3 Galois group decompositions of modules . . . 147
8.4 Scholz’s theorem . . . 151
8.5 Dutarte’s probabilistic model . . . 152
8.6 Some consequences . . . 154
9 Cyclic number fields 161 9.1 Introduction . . . 161
9.2 Group rings . . . 164
9.3 Number fields unramified at two . . . 168
9.4 Cubic and quintic number fields . . . 171
10 A large norm group 175 10.1 Main results . . . 175
10.2 Auxiliary results . . . 177
10.3 Proofs of the main results . . . 180
11 Quadratic characters 183 11.1 Main result . . . 183
11.2 Introduction . . . 184
11.3 L-series . . . 185
11.4 Lemmas . . . 186
11.5 Proof of the main theorem . . . 190
Bibliography 195
Summary 201
Samenvatting 203
Curriculum vitae 205
