Advanced Topics in the Arithmetic of Elliptic Curves

Joseph H. Silverman

ISBN 10: 0387943285 / ISBN 13: 9780387943282

Editore: Springer-Verlag New York Inc., US, 1994

Riguardo questo articolo

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. Softcover reprint of the original 1st ed. 1994.

Codice articolo LU-9780387943282

Dati bibliografici

• Titolo: Advanced Topics in the Arithmetic of Elliptic Curves
• Editore: Springer-Verlag New York Inc., US
• Data di pubblicazione: 1994
• Lingua: Inglese
• ISBN 10: 0387943285
• ISBN 13: 9780387943282
• Rilegatura: Paperback
• Condizione: New

Descrizione articolo

Riassunto

This book continues the treatment of the arithmetic theory of elliptic curves begun in the first volume. The book begins with the theory of elliptic and modular functions for the full modular group r(1), including a discussion of Hekcke operators and the L-series associated to cusp forms. This is followed by a detailed study of elliptic curves with complex multiplication, their associated Grossencharacters and L-series, and applications to the construction of abelian extensions of quadratic imaginary fields. Next comes a treatment of elliptic curves over function fields and elliptic surfaces, including specialization theorems for heights and sections. This material serves as a prelude to the theory of minimal models and Neron models of elliptic curves, with a discussion of special fibers, conductors, and Ogg's formula. Next comes a brief description of q-models for elliptic curves over C and R, followed by Tate's theory of q-models for elliptic curves with non-integral j-invariant over p-adic fields. The book concludes with the construction of canonical local height functions on elliptic curves, including explicit formulas for both archimedean and non-archimedean fields.

Contenuti

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.