ASPECTS OF COMPLEX MULTIPLICATION Contents

1. Preview 2 Complex multiplication on elliptic curves over C 2 Traces of singular moduli 3 Class field theory 3 The Kronecker limit formula and Kronecker’s solution of Pell’s equation 4 Application to Diophantine equations (Villegas) 4 L-series and CM modular forms 5 Other topics 6 2. Complex Multiplication on Elliptic Curves over C 6 Elliptic Curves over C 6 Elliptic functions 7 Complex multiplication 8 j is Algebraic 9 3. Complex Multiplication: A Modular Point of View 10 Theta series 10 Modular forms 12 Quadratic forms 13 Relation to CM points 14 The modular polynomial Ψ 14 A word on the polynomial Φ 15 Cyclic isogenies and the modular group 16 Calculation for n = 2 17 4. A Class Number Relation and Traces of Singular Moduli 18 Hurwitz-Kronecker class number relation 18 Traces of singular moduli: definitions 20 Traces of singular moduli: proofs I 22 Traces of singular moduli: proofs II 24 5. Constructing Class Fields 26 A review of class field theory 26 Kronecker’s congruence 27 The function φM (τ) and the polynomial Dn(X, j(τ)) 28 The proof of the congruence 30 Explicit examples 30 The values φM (τ) 31 The polynomial Gp(X,Y, Z) 32 A congruence property of ∆ 33 Summary 34