This book is meant to be an introductory text, albeit at an upper graduate level. The main prerequisite for reading this book is some familiarity with the basic theory of elliptic curves as described, for example, in the first volume. Numerous exercises have been included at the end of each chapter. A list of comments and citations for the exercises will be found at the end of the book.

Klappentext

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.

Inhalt

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