1. - I Elliptic and Modular Functions. - 1. The Modular Group. - 2. The Modular Curve X(1). - 3. Modular Functions. - 4. Uniformization and Fields of Moduli. - 5. Elliptic Functions Revisited. - 6. q-Expansions of Elliptic Functions. - 7. q-Expansions of Modular Functions. - 8. Jacobi's Product Formula for ?(?). - 9. Hecke Operators. - 10. Hecke Operators Acting on Modular Forms. - 11. L-Series Attached to Modular Forms. - Exercises. - II Complex Multiplication. - 1. Complex Multiplication over C. - 2. Rationality Questions. - 3. Class Field Theory A Brief Review. - 4. The Hilbert Class Field. - 5. The Maximal Abelian Extension. - 6. Integrality of j. - 7. Cyclotomic Class Field Theory. - 8. The Main Theorem of Complex Multiplication. - 9. The Associated Grössencharacter. - 10. The L-Series Attached to a CM Elliptic Curve. - Exercises. - III Elliptic Surfaces. - 1. Elliptic Curves over Function Fields. - 2. The Weak Mordell-Weil Theorem. - 3. Elliptic Surfaces. - 4. Heights on Elliptic Curves over Function Fields. - 5. Split Elliptic Surfaces and Sets of Bounded Height. - 6. The Mordell-Weil Theorem for Function Fields. - 7. The Geometry of Algebraic Surfaces. - 8. The Geometry of Fibered Surfaces. - 9. The Geometry of Elliptic Surfaces. - 10. Heights and Divisors on Varieties. - 11. Specialization Theorems for Elliptic Surfaces. - 12. Integral Points on Elliptic Curves over Function Fields. - Exercises. - IV The Néron Model. - 1. Group Varieties. - 2. Schemes and S-Schemes. - 3. Group Schemes. - 4. Arithmetic Surfaces. - 5. Néron Models. - 6. Existence of Néron Models. - 7. Intersection Theory Minimal Models and Blowing-Up. - 8. The Special Fiber of a Néron Model. - 9. Tate's Algorithm to Compute the Special Fiber. -10. The Conductor of an Elliptic Curve. - 11. Ogg's Formula. - Exercises. - V Elliptic Curves over Complete Fields. - 1. Elliptic Curves over ?. - 2. Elliptic Curves over ?. - 3. The Tate Curve. - 4. The Tate Map Is Surjective. - 5. Elliptic Curves over p-adic Fields. - 6. Some Applications of p-adic Uniformization. - Exercises. - VI Local Height Functions. - 1. Existence of Local Height Functions. - 2. Local Decomposition of the Canonical Height. - 3. Archimedean Absolute Values Explicit Formulas. - 4. Non-Archimedean Absolute Values Explicit Formulas. - Exercises. - Appendix A Some Useful Tables. - 3. Elliptic Curves over ? with Complex Multiplication. - Notes on Exercises. - References. - List of Notation. Language: English