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Complex Multiplication

Book Description

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Elliptic functions
    1. 1.1 Values of elliptic functions
    2. 1.2 The functions σ(z|L), ζ(z|L) and p(z|L)
    3. 1.3 Construction of elliptic functions
    4. 1.4 Algebraic and geometric properties of elliptic functions
    5. 1.5 Division polynomials
    6. 1.6 Weierstrass functions
      1. 1.6.1 Expansions at zero
      2. 1.6.2 p-adic limits
    7. 1.7 Elliptic resolvents
    8. 1.8 q-expansions
    9. 1.9 Dedekind’s η function and σ-product formula
    10. 1.10 The transformation formula of the Dedekind η function
  8. 2. Modular functions
    1. 2.1 The modular group
    2. 2.2 Congruence subgroups
    3. 2.3 Definition of modular forms
    4. 2.4 Examples of modular forms and modular functions
      1. 2.4.1 The functions g[sub(2)], g[sub(3)] and Δ
      2. 2.4.2 The functions j, 3[root(j)], 2[root(j − 12[sup(3)])], j[sub(R)], φ[sub(R)]
      3. 2.4.3 η-quotients
      4. 2.4.4 Weber’s τ function
      5. 2.4.5 The natural normalisation of the p function
      6. 2.4.6 Klein’s normalisation of the σ function
      7. 2.4.7 Transformation of τ[sup((e))], p, φ
    5. 2.5 Modular functions for Γ
      1. 2.5.1 Construction of modular functions for Γ
      2. 2.5.2 The q-expansion principle
    6. 2.6 Modular functions for subgroups of Γ
      1. 2.6.1 The isomorphisms of C[sub(U)]/C[sub(Γ)]
      2. 2.6.2 The extended q-expansion principle
    7. 2.7 Modular functions for Γ[sub(R)]
    8. 2.8 Modular functions for Γ(N)
    9. 2.9 The field Q(γ[sub(2)], γ[sub(3)])
    10. 2.10 Lower powers of η-quotients
  9. 3. Basic facts from number theory
    1. 3.1 Ideal theory of suborders in a quadratic number field
      1. 3.1.1 Fractional ideals, integral ideals, proper ideals, regular ideals
      2. 3.1.2 Ideal groups
      3. 3.1.3 Primitive matrices and bases of ideals
      4. 3.1.4 Integral ideals that are not regular
    2. 3.2 Density theorems
    3. 3.3 Class field theory
  10. 4. Factorisation of singular values
    1. 4.1 Singular values
    2. 4.2 Factorisation of φ[sub(A)](α)
    3. 4.3 Factorisation of φ(ξ | L)
    4. 4.4 A result of Dorman, Gross and Zagier
  11. 5. The Reciprocity Law
    1. 5.1 The Reciprocity Law of Weber, Hasse, Söhngen, Shimura
    2. 5.2 Applications of the Reciprocity Law
  12. 6. Generation of ring class fields and ray class fields
    1. 6.1 Generation of ring class fields by singular values of j
    2. 6.2 Generation of ray class fields by τ and j
    3. 6.3 The singular values of γ[sub(2)] and γ[sub(3)]
    4. 6.4 The singular values of Schläfli’s functions
    5. 6.5 Heegner’s solution of the class number one problem
    6. 6.6 Generation of ring class fields by η-quotients
    7. 6.7 Double η-quotients in the ramified case
    8. 6.8 Generation of ray class fields by φ(z|[stacked[ω[sub(1)]][ω[sub(2)]]])
    9. 6.9 Generalised principal ideal theorem
  13. 7. Integral basis in ray class fields
    1. 7.1 A normalisation of the Weierstrass p function
    2. 7.2 The discriminant of P(δ)
    3. 7.3 The denominator of P(δ)
    4. 7.4 Construction of relative integral basis
      1. 7.4.1 Analogy to cyclotomic fields
    5. 7.5 Relative integral power basis
    6. 7.6 Bley’s generalisation for K[sub(t,f)]/Ω[sub(t)] with t > 1
  14. 8. Galois module structure
    1. 8.1 Torsion points and good reduction
    2. 8.2 Kummer theory of E
    3. 8.3 Integral objects
    4. 8.4 Global construction of O[sub(P)] and A as O[sub(L)]-algebras
    5. 8.5 Construction of a generating element for O[sub(P)] over A
    6. 8.6 Galois module structure of ray class fields
    7. 8.7 Models of elliptic curves
      1. 8.7.1 The Weierstrass model
      2. 8.7.2 The Fueter model
      3. 8.7.3 The Deuring model
      4. 8.7.4 Singular values of the Weierstrass, Fueter and Deuring functions
      5. 8.7.5 Singular values of Weierstrass functions
    8. 8.8 Proofs of Theorems 8.3.1 and 8.5.1
    9. 8.9 Proofs of Theorems 8.4.1, 8.4.2 and 8.5.2
    10. 8.10 Proofs of Theorems 8.9.2 and 8.6.2
    11. 8.11 Analogy to the cyclotomic case
    12. 8.12 Generalisation to ring classes by Bettner and Bley
  15. 9. Berwick’s congruences
    1. 9.1 Bettner’s results
    2. 9.2 Method of proof
  16. 10. Cryptographically relevant elliptic curves
    1. 10.1 Reduction of the Weierstrass model
    2. 10.2 Computation of j(O) modulo P
      1. 10.2.1 Schläfli–Weber functions
      2. 10.2.2 Double η-quotients
      3. 10.2.3 Application of η-quotients in the ramified case
    3. 10.3 Reduction of the Fueter and Deuring models
      1. 10.3.1 Reduction of the Fueter model
      2. 10.3.2 Reduction of the Deuring model
  17. 11. The class number formulae of Curt Meyer
    1. 11.1 L-Functions of ring class characters
    2. 11.2 L-function s of ray class characters χ with f[sub(χ)] ≠ (1).
    3. 11.3 Class number formulae
  18. 12. Arithmetic interpretation of class number formulae
    1. 12.1 Group-theoretical lemmas for the case L ⊇ K
    2. 12.2 Applications of Theorems 12.1.1, 12.1.2
      1. 12.2.1 Application of Theorem 12.1.1
      2. 12.2.2 Application of Theorem 12.1.2
    3. 12.3 Class number formulae for Ω ⊇ L ⊇ K
    4. 12.4 Class number formulae for K[sub(f)] ⊇ L ⊇ K
      1. 12.4.1 Application of the formulae from 12.4
    5. 12.5 Group-theoretical lemmas for M ⊉ K
    6. 12.6 The Galois group of MK/K
    7. 12.7 Class number formulae for Ω ⊃ M ⊉ K
    8. 12.8 Class number formulae for K[sub(f)] ⊃ M ⊅[(≠) below] K
      1. 12.8.1 Applications of the class number formulae in 12.8
  19. References
  20. Index of Notation
  21. Index