One General Theory.- 1 Elliptic Functions.- 2 Homomorphisms.- 3 The Modular Function.- 4 Fourier Expansions.- 5 The Modular Equation.- 6 Higher Levels.- 7 Automorphisms of the Modular Function Field.- Two Complex Multiplication Elliptic Curves With Singular Invariants.- 8 Results from Algebraic Number Theory.- 9 Reduction of Elliptic Curves.- 10 Complex Multiplication.- 11 Shimura's Reciprocity Law.- 12 The Function ?(??)/?(?).- 13 The ?-adic and p-adic Representations of Deuring.- 14 Ihara's Theory.- Three Elliptic Curves with Non-Integral Invariant.- 15 The Tate Parametrization.- 16 The Isogeny Theorems.- 17 Division Points Over Number Fields.- Four Theta Functions and Kronecker Limit Formula.- 18 Product Expansions.- 19 The Siegel Functions and Klein Forms.- 20 The Kronecker Limit Formulas.- 21 The First Limit Formula and L-series.- 22 The Second Limit Formula and L-series.- Appendix 1 Algebraic Formulas in Arbitrary Characteristic.- By J. Tate.- 1 Generalized Weierstrass Form.- 2 Canonical Forms.- Appendix 2 The Trace of Frobenius and the Differential of First Kind.- 1 The Trace of Frobenius.- 2 Duality.- 3 The Tate Trace.- 4 The Cartier Operator.- 5 The Hasse Invariant.