Author	Silverman, Joseph H. author
Title	Advanced Topics in the Arithmetic of Elliptic Curves [electronic resource] / by Joseph H. Silverman
Imprint	New York, NY : Springer New York : Imprint: Springer, 1994
Connect to	http://dx.doi.org/10.1007/978-1-4612-0851-8
Descript	XIII, 528 p. online resource

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 -- ยง1. The Modular Group -- ยง2. The Modular Curve X(1) -- ยง3. Modular Functions -- ยง4. Uniformization and Fields of Moduli -- ยง5. Elliptic Functions Revisited -- ยง6. q-Expansions of Elliptic Functions -- ยง7. q-Expansions of Modular Functions -- ยง8. Jacobiโ{128}{153}s Product Formula for ?(?) -- ยง9. Hecke Operators -- ยง10. Hecke Operators Acting on Modular Forms -- ยง11. L-Series Attached to Modular Forms -- Exercises -- II Complex Multiplication -- ยง1. Complex Multiplication over C -- ยง2. Rationality Questions -- ยง3. Class Field Theory โ{128}{148} A Brief Review -- ยง4. The Hilbert Class Field -- ยง5. The Maximal Abelian Extension -- ยง6. Integrality of j -- ยง7. Cyclotomic Class Field Theory -- ยง8. The Main Theorem of Complex Multiplication -- ยง9. The Associated Grรถssencharacter -- ยง10. The L-Series Attached to a CM Elliptic Curve -- Exercises -- III Elliptic Surfaces -- ยง1. Elliptic Curves over Function Fields -- ยง2. The Weak Mordell-Weil Theorem -- ยง3. Elliptic Surfaces -- ยง4. Heights on Elliptic Curves over Function Fields -- ยง5. Split Elliptic Surfaces and Sets of Bounded Height -- ยง6. The Mordell-Weil Theorem for Function Fields -- ยง7. The Geometry of Algebraic Surfaces -- ยง8. The Geometry of Fibered Surfaces -- ยง9. The Geometry of Elliptic Surfaces -- ยง10. Heights and Divisors on Varieties -- ยง11. Specialization Theorems for Elliptic Surfaces -- ยง12. Integral Points on Elliptic Curves over Function Fields -- Exercises -- IV The Nรฉron Model -- ยง1. Group Varieties -- ยง2. Schemes and S-Schemes -- ยง3. Group Schemes -- ยง4. Arithmetic Surfaces -- ยง5. Nรฉron Models -- ยง6. Existence of Nรฉron Models -- ยง7. Intersection Theory, Minimal Models, and Blowing-Up -- ยง8. The Special Fiber of a Nรฉron Model -- ยง9. Tateโ{128}{153}s Algorithm to Compute the Special Fiber -- ยง10. The Conductor of an Elliptic Curve -- ยง11. Oggโ{128}{153}s Formula -- Exercises -- V Elliptic Curves over Complete Fields -- ยง1. Elliptic Curves over ? -- ยง2. Elliptic Curves over ? -- ยง3. The Tate Curve -- ยง4. The Tate Map Is Surjective -- ยง5. Elliptic Curves over p-adic Fields -- ยง6. Some Applications of p-adic Uniformization -- Exercises -- VI Local Height Functions -- ยง1. Existence of Local Height Functions -- ยง2. Local Decomposition of the Canonical Height -- ยง3. Archimedean Absolute Values โ{128}{148} Explicit Formulas -- ยง4. Non-Archimedean Absolute Values โ{128}{148} Explicit Formulas -- Exercises -- Appendix A Some Useful Tables -- ยง3. Elliptic Curves over ? with Complex Multiplication -- Notes on Exercises -- References -- List of Notation