目录
 前辅文
 Chapter 1 Background: history, conjectures and theorems
  1.1 Elliptic curves over rationals and finite fields
  1.2 Sato–Tate and Lang–Trotter conjectures
  1.3 Our main results
 Chapter 2 The Hardy–Littlewood conjecture and upper bound for πE,r(x)
  2.1 The Hardy–Littlewood conjecture and sieve methods
  2.2 Elliptic curves with complex multiplication
  2.3 Proof of Theorem 1.3
  2.4 Fixed trace in imaginary quadratic fields
 Chapter 3 Power residues and laws of reciprocity
  3.1 Quadratic residues and reciprocity
  3.2 Quartic residues and reciprocity
  3.3 An incomplete character sum and quartic Gauß sums
  3.4 Cubic residues and reciprocity
  3.5 An incomplete character sum and cubic Gauß sums
  3.6 Remarks
 Chapter 4 Complex multiplication by Q(-1)
  4.1 Asymptotics for πE,r(x): conditional results
  4.2 Initial transformations
  4.3 Asymptotic evaluation of Gd(x; r, α, β, γ)
  4.4 From Conjecture 4.3 to Lang–Trotter
 Chapter 5 Complex multiplication by Q(-3)
  5.1 Asymptotics for πE,r(x): conditional results
  5.2 Initial transformations
  5.3 Asymptotic evaluation of Ed,k,ϵ(x; r, α, β, γ, η)
  5.4 From Conjecture 5.5 to Lang–Trotter
 Chapter 6 Complex multiplication by remaining imaginary quadratic fields
  6.1 Asymptotics for πE,r(x): conditional results
  6.2 Initial transformations
  6.3 Asymptotic evaluation of ΠD(x; r, q, a)
  6.4 From Conjecture 6.8 to Lang–Trotter: D = 2
  6.5 From Conjecture 6.8 to Lang–Trotter: D ≥ 7
 Chapter 7 Further interpretations on the Lang–Trotter constant
  7.1 Positivity
  7.2 Anomalous primes
  7.3 Symmetry
  7.4 Image of Galois representations
 Chapter 8 Historical notes and some perspectives
  8.1 Gauß’ last diary entry and Disquisitiones Arithmeticae
  8.2 Variants and generalizations of Lang–Trotter conjecture
 Chapter 9 Numerical data
  9.1 CM by Q(-1)
  9.2 CM by Q(-3)
  9.3 CM by Q(-7)
 Bibliography
 Appendix A Evaluation of ϖE, r, with CM by Q(-3)
  A.1 r≡±4 (mod 24)
  A.2 r≡±10 (mod 12)
  A.3 r≡±1 (mod 6)
 前辅文
 Chapter 1 Background: history, conjectures and theorems
  1.1 Elliptic curves over rationals and finite fields
  1.2 Sato–Tate and Lang–Trotter conjectures
  1.3 Our main results
 Chapter 2 The Hardy–Littlewood conjecture and upper bound for πE,r(x)
  2.1 The Hardy–Littlewood conjecture and sieve methods
  2.2 Elliptic curves with complex multiplication
  2.3 Proof of Theorem 1.3
  2.4 Fixed trace in imaginary quadratic fields
 Chapter 3 Power residues and laws of reciprocity
  3.1 Quadratic residues and reciprocity
  3.2 Quartic residues and reciprocity
  3.3 An incomplete character sum and quartic Gauß sums
  3.4 Cubic residues and reciprocity
  3.5 An incomplete character sum and cubic Gauß sums
  3.6 Remarks
 Chapter 4 Complex multiplication by Q(-1)
  4.1 Asymptotics for πE,r(x): conditional results
  4.2 Initial transformations
  4.3 Asymptotic evaluation of Gd(x; r, α, β, γ)
  4.4 From Conjecture 4.3 to Lang–Trotter
 Chapter 5 Complex multiplication by Q(-3)
  5.1 Asymptotics for πE,r(x): conditional results
  5.2 Initial transformations
  5.3 Asymptotic evaluation of Ed,k,ϵ(x; r, α, β, γ, η)
  5.4 From Conjecture 5.5 to Lang–Trotter
 Chapter 6 Complex multiplication by remaining imaginary quadratic fields
  6.1 Asymptotics for πE,r(x): conditional results
  6.2 Initial transformations
  6.3 Asymptotic evaluation of ΠD(x; r, q, a)
  6.4 From Conjecture 6.8 to Lang–Trotter: D = 2
  6.5 From Conjecture 6.8 to Lang–Trotter: D ≥ 7
 Chapter 7 Further interpretations on the Lang–Trotter constant
  7.1 Positivity
  7.2 Anomalous primes
  7.3 Symmetry
  7.4 Image of Galois representations
 Chapter 8 Historical notes and some perspectives
  8.1 Gauß’ last diary entry and Disquisitiones Arithmeticae
  8.2 Variants and generalizations of Lang–Trotter conjecture
 Chapter 9 Numerical data
  9.1 CM by Q(-1)
  9.2 CM by Q(-3)
  9.3 CM by Q(-7)
 Bibliography
 Appendix A Evaluation of ϖE, r, with CM by Q(-3)
  A.1 r≡±4 (mod 24)
  A.2 r≡±10 (mod 12)
  A.3 r≡±1 (mod 6)