Advanced Topics in the Arithmetic of Elliptic Curves

Produktbeschreibung

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. 

Inhaltsverzeichnis

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 N‚ron Model.-
1. Group Varieties.-
2. Schemes and S-Schemes.-
3. Group Schemes.-
4. Arithmetic Surfaces.-
5. N‚ron Models.-
6. Existence of N‚ron Models.-
7. Intersection Theory, Minimal Models, and Blowing-Up.-
8. The Special Fiber of a N‚ron 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.