Advanced Topics in the Arithmetic of Elliptic Curves

✍ Scribed by Joseph H. Silverman (auth.)

• Publisher: Springer-Verlag New York
• Year: 1994
• Tongue: English
• Leaves: 540
• Series: Graduate Texts in Mathematics 151
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Contents
Introduction
Chapter I: Elliptic and Modular Functions
I.1: The Modular Group
I.2: The Modular Curve X(1)
I.3: Modular Functions
I.4: Uniformization and Fields of Moduli
I.5: Elliptic Functions Revisited
I.6: q-Expansions of Elliptic Functions
I.7: q-Expansions of Modular Functions
I.8: Jacobi's Product Formula for Δ(τ)
I.9: Hecke Operators
I.10: Hecke Operators Acting on Modular Forms
I.11: L-Series Attached to Modular Forms
Exercises
Chapter II: Complex Multiplication
II.1: Complex Multiplication over ℂ
II.2: Rationality Questions
II.3: Class Field Theory — A Brief Review
II.4: The Hilbert Class Field
II.5: The Maximal Abelian Extension
II.6: Integrality of j
II.7: Cyclotomic Class Field Theory
II.8: The Main Theorem of Complex Multiplication
II.9: The Associated Grössencharacter
II.10: The L-Series Attached to a CM Elliptic Curve
Exercises
Chapter III: Elliptic Surfaces
III.1: Elliptic Curves over Function Fields
III.2: The Weak Mordell-Weil Theorem
III.3: Elliptic Surfaces
III.4: Heights on Elliptic Curves over Function Fields
III.5: Split Elliptic Surfaces and Sets of Bounded Height
III.6: The Mordell-Weil Theorem for Function Fields
III.7: The Geometry of Algebraic Surfaces
III.8: The Geometry of Fibered Surfaces
III.9: The Geometry of Elliptic Surfaces
III.10: Heights and Divisors on Varieties
III.11: Specialization Theorems for Elliptic Surfaces
III.12: Integral Points on Elliptic Curves over Function Fields
Exercises
Chapter IV: The Néron Model
IV.1: Group Varieties
IV.2: Schemes and S-Schemes
IV.3: Group Schemes
IV.4: Arithmetic Surfaces
IV.5: Néron Models
IV.6: Existence of Néron Models
IV.7: Intersection Theory, Minimal Models, and Blowing-Up
IV.8: The Special Fiber of a Néron Model
IV.9: Tate's Algorithm to Compute the Special Fiber
IV.10: The Conductor of an Elliptic Curve
IV.11: Ogg's Formula
Exercises
Chapter V: Elliptic Curves over Complete Fields
V.1: Elliptic Curves over ℂ
V.2: Elliptic Curves over ℝ
V.3: The Tate Curve
V.4: The Tate Map Is Surjective
V.5: Elliptic Curves over p-adic Fields
V.6: Some Applications of p-adic Uniformization
Exercises
Chapter VI: Local Height Functions
VI.1: Existence of Local Height Functions
VI.2: Local Decomposition of the Canonical Height
VI.3: Archimedean Absolute Values — Explicit Formulas
VI.4: Non-Archimedean Absolute Values — Explicit Formulas
Exercises
Appendix A: Some Useful Tables
A.1: Bernoulli Numbers and ζ(2k)
A.2: Fourier Coefficients of Δ(τ) and j(τ)
A.3: Elliptic Curves over ℚ with Complex Multiplication
Notes on Exercises
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
References
List of Notation
Index