Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Elliptic functions -- 1.1 Values of elliptic functions -- 1.2 The functions… -- 1.3 Construction of elliptic functions -- 1.4 Algebraic and geometric properties of elliptic functions -- 1.5 Division polynomials -- 1.6 Weierstrass functions -- 1.6.1 Expansions at zero -- 1.6.2 p-adic limits -- 1.7 Elliptic resolvents -- 1.8 q-expansions -- 1.9 Dedekind's eta function and sigma-product formula -- 1.10 The transformation formula of the Dedekind eta function -- 2 Modular functions -- 2.1 The modular group -- 2.2 Congruence subgroups -- 2.3 Definition of modular forms -- 2.4 Examples of modular forms and modular functions -- 2.4.1 The functions g2,g3 and delta -- 2.4.2 The functions… -- 2.4.3 eta-quotients -- 2.4.4 Weber's tau function -- 2.4.5 The natural normalisation of the… -- 2.4.6 Klein's normalisation of the sigma function -- 2.4.7 Transformation of… -- 2.5 Modular functions for Gamma -- 2.5.1 Construction of modular functions for Gamma -- 2.5.2 The q-expansion principle -- 2.6 Modular functions for subgroups of Gamma -- 2.6.1 The isomorphisms of CU/CGamma -- 2.6.2 The extended q-expansion principle -- 2.7 Modular functions for GammaR -- 2.8 Modular functions for Gamma(N) -- 2.9 The field Q(gamma2, gamma3) -- 2.10 Lower powers of eta-quotients -- 3 Basic facts from number theory -- 3.1 Ideal theory of suborders in a quadratic number field -- 3.1.1 Fractional ideals, integral ideals, proper ideals, regular ideals -- 3.1.2 Ideal groups -- 3.1.3 Primitive matrices and bases of ideals -- 3.1.4 Integral ideals that are not regular -- 3.2 Density theorems -- 3.3 Class field theory -- 4 Factorisation of singular values -- 4.1 Singular values -- 4.2 Factorisation of varphi A (alpha) -- 4.3 Factorisation of… -- 4.4 A result of Dorman, Gross and Zagier.

5 The Reciprocity Law -- 5.1 The Reciprocity Law of Weber, Hasse, Söhngen, Shimura -- 5.2 Applications of the Reciprocity Law -- 6 Generation of ring class fields and ray class fields -- 6.1 Generation of ring class fields by singular values of j -- 6.2 Generation of ray class fields by tau and j -- 6.3 The singular values of gamma 2 and gamma 3 -- 6.4 The singular values of Schläfli's functions -- 6.5 Heegner's solution of the class number one problem -- 6.6 Generation of ring class fields by eta-quotients -- 6.7 Double eta-quotients in the ramified case -- 6.8 Generation of ray class fields by… -- 6.9 Generalised principal ideal theorem -- 7 Integral basis in ray class fields -- 7.1 A normalisation of the Weierstrass… -- 7.2 The discriminant of… -- 7.3 The denominator of… -- 7.4 Construction of relative integral basis -- 7.4.1 Analogy to cyclotomic fields -- 7.5 Relative integral power basis -- 7.6 Bley's generalisation for… -- 8 Galois module structure -- 8.1 Torsion points and good reduction -- 8.2 Kummer theory of E -- 8.3 Integral objects -- 8.4 Global construction of… -- 8.5 Construction of a generating element for… -- 8.6 Galois module structure of ray class fields -- 8.7 Models of elliptic curves -- 8.7.1 The Weierstrass model -- 8.7.2 The Fueter model -- 8.7.3 The Deuring model -- 8.7.4 Singular values of the Weierstrass, Fueter and Deuring functions -- 8.7.5 Singular values of Weierstrass functions -- 8.8 Proofs of Theorems 8.3.1 and 8.5.1 -- 8.9 Proofs of Theorems 8.4.1, 8.4.2 and 8.5.2 -- 8.10 Proofs of Theorems 8.9.2 and 8.6.2 -- 8.11 Analogy to the cyclotomic case -- 8.12 Generalisation to ring classes by Bettner and Bley -- 9 Berwick's congruences -- 9.1 Bettner's results -- 9.2 Method of proof -- 10 Cryptographically relevant elliptic curves -- 10.1 Reduction of the Weierstrass model -- 10.2 Computation of….

10.2.1 Schläfli-Weber functions -- 10.2.2 Double eta-quotients -- 10.2.3 Application of eta-quotients in the ramified case -- 10.3 Reduction of the Fueter and Deuring models -- 10.3.1 Reduction of the Fueter model -- 10.3.2 Reduction of the Deuring model -- 11 The class number formulae of Curt Meyer -- 11.1 L-Functions of ring class characters -- 11.2 L-function s of ray class characters… -- 11.3 Class number formulae -- 12 Arithmetic interpretation of class number formulae -- 12.1 Group-theoretical lemmas for the case… -- 12.2 Applications of Theorems 12.1.1, 12.1.2 -- 12.2.1 Application of Theorem 12.1.1 -- 12.2.2 Application of Theorem 12.1.2 -- 12.3 Class number formulae for… -- 12.4 Class number formulae for… -- 12.4.1 Application of the formulae from 12.4 -- 12.4.1.1 Divisibility between class numbers -- 12.4.1.2 Divisibility of class numbers by divisors of the field degree -- 12.5 Group-theoretical lemmas for… -- 12.6 The Galois group of MK/K -- 12.7 Class number formulae for… -- 12.8 Class number formulae for… -- 12.8.1 Applications of the class number formulaein 12.8 -- 12.8.1.1 Divisibility relations between class numbers -- 12.8.1.2 Divisibility of class numbers by divisors of the field degree -- References -- Index of Notation -- Index.