Advanced Topics in the Arithmetic of Elliptic Curves

In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.

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.