| Dewey Class |
516.35 |
| Title |
Advanced Topics in the Arithmetic of Elliptic Curves ([EBook]) / by Joseph H. Silverman. |
| Author |
Silverman, Joseph H. , 1955- |
| Other name(s) |
SpringerLink (Online service) |
| Publication |
New York, NY : Springer , 1994. |
| Physical Details |
XIII, 528 p. : online resource. |
| Series |
Graduate texts in mathematics 0072-5285 ; ; 151 |
| ISBN |
9781461208518 |
| Summary Note |
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.: |
| Contents note |
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. |
| System details note |
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users) |
| Internet Site |
http://dx.doi.org/10.1007/978-1-4612-0851-8 |
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