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.