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. Language: English