Explicit Brauer Induction: With Applications to Algebra and Number Theory

Contents......Page 5
Preface......Page 7
1.1 Basic definitions......Page 13
1.2 Complex representations......Page 17
1.3 Exercises......Page 33
2 Induction theorems......Page 35
2.1 Induction theorems of Artin and Brauer......Page 37
2.2 Brauer induction in canonical rational form......Page 44
2.3 Brauer induction in canonical integral form......Page 57
2.4 Inductive explicit Brauer induction......Page 66
2.5 Exercises......Page 79
3 GL_2F_q......Page 84
3.1 Weil representations......Page 85
3.2 Explicit Brauer induction and Shintani descent......Page 101
3.3 Exercises......Page 116
4 The class-group of a group-ring......Page 118
4.1 Adams operations and rationality......Page 119
4.2 Describing the class-group by representations......Page 122
4.3 Determinantal congruences......Page 132
4.4 Detecting elements in the class-group......Page 143
4.5 Galois properties of local determinants......Page 150
4.6 Adams operations and determinants......Page 165
4.7 Exercises......Page 178
5 A class-group miscellany......Page 182
5.1 Restricted determinants......Page 184
5.2 The class-group of Z[Qs]......Page 188
5.3 Relations between Swan modules......Page 201
5.4 The class-group of a maximal order......Page 217
5.5 Swan subgroups for nilpotent groups......Page 231
5.6 Cyclic groups......Page 242
5.7 Exercises......Page 253
6 Complete discrete valuation fields......Page 257
6.1 Ramification groups and functions......Page 258
6.2 Kato's abelian conductor......Page 270
6.3 The non-abelian Swan conductor......Page 294
6.4 Exercises......Page 309
7 Galois module structure......Page 311
7.1 Local Chinburg invariants......Page 312
7.2 The global Chinburg invariant......Page 343
7.3 The Chinburg invariant modulo D(Z[G])......Page 349
7.4 Real cyclotomic Galois module structure......Page 377
7.5 Exercises......Page 408
Bibliography......Page 415
Index......Page 419