Advanced topics in the arithmetic of elliptic curves

著者

• Silverman, Joseph H.

書誌事項

Advanced topics in the arithmetic of elliptic curves

Joseph H. Silverman

（Graduate texts in mathematics, 151）

Springer-Verlag, c1994

• : us : hard
• : us : soft
• : gw : hard
• : gw : soft

注記

Includes bibliographical references (p. [488]-497) and index

内容説明・目次

巻冊次

: us : hard ISBN 9780387943251

内容説明

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.

目次

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.

巻冊次

: us : soft ISBN 9780387943282

内容説明

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.

目次

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.

巻冊次

: gw : hard ISBN 9783540943259

内容説明

In The Arithmetic of Elliptic Curves, the author presented basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. This book continues the study of elliptic curves by presenting six important, but somewhat more specialised topics: I. Elliptic and modular functions for the full modular group. Il. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialisation theorems. IV. Neron models, Kodaira-N ron classification of special fibres, 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.

巻冊次

: gw : soft ISBN 9783540943280

内容説明

In the first volume of The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. This second volume continues the study of elliptic curves by presenting six important, somewhat more specialized topics: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. N ron models, Kodaira-N ron classification of special fibres, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. N ron's theory of canonical local height functions.

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詳細情報

• NII書誌ID(NCID): BA23879440
• ISBN:
  • 0387943250
  • 0387943285
  • 3540943250
  • 3540943285
• LCCN: 94021787
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: New York ; Tokyo
• ページ数/冊数: xiii, 525 p.
• 大きさ: 25 cm
• 分類:
  • LCC : QA567
  • DC20 : 516.3/52
  • NDC8 : 411.8
• 件名:
  • LCSH : Curves, Elliptic
  • LCSH : Curves, Algebraic
  • LCSH : Arithmetic
• 親書誌ID:
  • BA00010074