Advanced Topics in the Arithmetic of Elliptic Curves

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.