Lang--Trotter Conjecture for CM Elliptic Curves

Abstract

Notation vii Chapter 1. Backgrounds: 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 π E,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 π E,r (x): conditional results 4.2. Initial transformations 4.3. Asymptotic evaluation of G d (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 π E,r (x): conditional results 5.2. Initial transformations 5.3. Asymptotic evaluation of E d,k,ǫ (x; r, α, β, γ) 5.4. From Conjecture 5.5 to Lang-Trotter Chapter 6. Complex multiplication by remaining imaginary quadratic fields 6.1. Asymptotics for π E,r (x): conditional results 6.2. Initial transformations 6.3. Asymptotic evaluation of Π D (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 iii iv CONTENTS 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 Bibliography