This is a self-contained account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers.
Author(s): Reinhard Schertz
Series: New Mathematical Monographs
Publisher: CUP
Year: 2010
Language: English
Pages: 377
Contents......Page 9
Preface......Page 13
1.1 Values of elliptic functions......Page 17
1.2 The functions…......Page 19
1.3 Construction of elliptic functions......Page 23
1.4 Algebraic and geometric properties of elliptic functions......Page 25
1.5 Division polynomials......Page 29
1.6 Weierstrass functions......Page 32
1.6.1 Expansions at zero......Page 34
1.6.2 p-adic limits......Page 39
1.7 Elliptic resolvents......Page 43
1.8 q-expansions......Page 48
1.9 Dedekind's eta function and sigma-product formula......Page 51
1.10 The transformation formula of the Dedekind eta function......Page 54
2 Modular functions......Page 57
2.1 The modular group......Page 58
2.2 Congruence subgroups......Page 61
2.3 Definition of modular forms......Page 64
2.4.2 The functions…......Page 66
2.4.3 eta-quotients......Page 67
2.4.4 Weber's tau function......Page 68
2.4.7 Transformation of…......Page 69
2.5.1 Construction of modular functions for Gamma......Page 70
2.5.2 The q-expansion principle......Page 75
2.6.1 The isomorphisms of CU/CGamma......Page 77
2.6.2 The extended q-expansion principle......Page 78
2.7 Modular functions for GammaR......Page 79
2.8 Modular functions for Gamma(N)......Page 85
2.9 The field Q(gamma2, gamma3)......Page 88
2.10 Lower powers of eta-quotients......Page 90
3.1.1 Fractional ideals, integral ideals, proper ideals, regular ideals......Page 98
3.1.2 Ideal groups......Page 102
3.1.3 Primitive matrices and bases of ideals......Page 110
3.1.4 Integral ideals that are not regular......Page 114
3.2 Density theorems......Page 116
3.3 Class field theory......Page 119
4.1 Singular values......Page 127
4.2 Factorisation of varphi A (alpha)......Page 130
4.3 Factorisation of…......Page 134
4.4 A result of Dorman, Gross and Zagier......Page 137
5.1 The Reciprocity Law of Weber, Hasse, Söhngen, Shimura......Page 138
5.2 Applications of the Reciprocity Law......Page 144
6.1 Generation of ring class fields by singular values of j......Page 154
6.2 Generation of ray class fields by tau and j......Page 157
6.3 The singular values of gamma 2 and gamma 3......Page 160
6.4 The singular values of Schläfli's functions......Page 164
6.5 Heegner's solution of the class number one problem......Page 167
6.6 Generation of ring class fields by eta-quotients......Page 170
6.7 Double eta-quotients in the ramified case......Page 181
6.8 Generation of ray class fields by…......Page 185
6.9 Generalised principal ideal theorem......Page 199
7 Integral basis in ray class fields......Page 206
7.1 A normalisation of the Weierstrass…......Page 207
7.2 The discriminant of…......Page 209
7.3 The denominator of…......Page 213
7.4 Construction of relative integral basis......Page 217
7.4.1 Analogy to cyclotomic fields......Page 219
7.5 Relative integral power basis......Page 221
7.6 Bley's generalisation for…......Page 226
8 Galois module structure......Page 229
8.1 Torsion points and good reduction......Page 230
8.2 Kummer theory of E......Page 231
8.3 Integral objects......Page 233
8.4 Global construction of…......Page 236
8.5 Construction of a generating element for…......Page 237
8.6 Galois module structure of ray class fields......Page 240
8.7.1 The Weierstrass model......Page 244
8.7.2 The Fueter model......Page 245
8.7.3 The Deuring model......Page 247
8.7.4 Singular values of the Weierstrass, Fueter and Deuring functions......Page 248
8.7.5 Singular values of Weierstrass functions......Page 250
8.8 Proofs of Theorems 8.3.1 and 8.5.1......Page 254
8.9 Proofs of Theorems 8.4.1, 8.4.2 and 8.5.2......Page 261
8.10 Proofs of Theorems 8.9.2 and 8.6.2......Page 266
8.11 Analogy to the cyclotomic case......Page 269
8.12 Generalisation to ring classes by Bettner and Bley......Page 272
9.1 Bettner's results......Page 277
9.2 Method of proof......Page 279
10.1 Reduction of the Weierstrass model......Page 282
10.2 Computation of…......Page 289
10.2.1 Schläfli–Weber functions......Page 291
10.2.2 Double eta-quotients......Page 292
10.2.3 Application of eta-quotients in the ramified case......Page 294
10.3.1 Reduction of the Fueter model......Page 298
10.3.2 Reduction of the Deuring model......Page 301
11 The class number formulae of Curt Meyer......Page 304
11.1 L-Functions of ring class characters......Page 305
11.2 L-function s of ray class characters…......Page 307
11.3 Class number formulae......Page 309
12.1 Group-theoretical lemmas for the case…......Page 311
12.2 Applications of Theorems 12.1.1, 12.1.2......Page 317
12.2.1 Application of Theorem 12.1.1......Page 318
12.2.2 Application of Theorem 12.1.2......Page 319
12.3 Class number formulae for…......Page 320
12.4 Class number formulae for…......Page 325
12.4.1.1 Divisibility between class numbers......Page 333
12.4.1.2 Divisibility of class numbers by divisors of the field degree......Page 338
12.5 Group-theoretical lemmas for…......Page 339
12.6 The Galois group of MK/K......Page 352
12.7 Class number formulae for…......Page 354
12.8 Class number formulae for…......Page 357
12.8.1.1 Divisibility relations between class numbers......Page 362
12.8.1.2 Divisibility of class numbers by divisors of the field degree......Page 364
References......Page 367
Index of Notation......Page 372
Index......Page 376
