Description
Springer Advanced Topics in the Arithmetic of Elliptic Curves 1999 Edition by Joseph H. Silverman
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. Table of contents : - 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 Groessencharacter.- 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 Neron Model.- 1. Group Varieties.- 2. Schemes and S-Schemes.- 3. Group Schemes.- 4. Arithmetic Surfaces.- 5. Neron Models.- 6. Existence of Neron Models.- 7. Intersection Theory, Minimal Models, and Blowing-Up.- 8. The Special Fiber of a Neron 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.