Advanced Topics in the Arithmetic of Elliptic Curves, Volume 151In 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. |
Contents
Introduction | 1 |
2 The Modular Curve X1 | 14 |
3 Modular Functions | 23 |
4 Uniformization and Fields of Moduli | 34 |
6 qExpansions of Elliptic Functions | 47 |
7 qExpansions of Modular Functions | 55 |
8 Jacobis Product Formula for τ | 62 |
10 Hecke Operators Acting on Modular Forms | 74 |
11 Specialization Theorems for Elliptic Surfaces | 265 |
12 Integral Points on Elliptic Curves over Function Fields | 274 |
Exercises | 278 |
CHAPTER IV | 289 |
1 Group Varieties | 290 |
2 Schemes and SSchemes | 297 |
3 Group Schemes | 306 |
4 Arithmetic Surfaces | 311 |
CHAPTER II | 95 |
2 Rationality Questions | 104 |
3 Class Field Theory A Brief Review | 115 |
4 The Hilbert Class Field | 121 |
5 The Maximal Abelian Extension | 128 |
6 Integrality of j | 140 |
7 Cyclotomic Class Field Theory | 151 |
8 The Main Theorem of Complex Multiplication | 157 |
9 The Associated Grössencharacter | 165 |
10 The LSeries Attached to a CM Elliptic Curve | 171 |
Exercises | 178 |
Elliptic Surfaces | 187 |
1 Elliptic Curves over Function Fields | 188 |
2 The Weak MordellWeil Theorem | 191 |
3 Elliptic Surfaces | 200 |
4 Heights on Elliptic Curves over Function Fields | 212 |
5 Split Elliptic Surfaces and Sets of Bounded Height | 220 |
6 The MordellWeil Theorem for Function Fields | 230 |
7 The Geometry of Algebraic Surfaces | 231 |
8 The Geometry of Fibered Surfaces | 236 |
9 The Geometry of Elliptic Surfaces | 245 |
10 Heights and Divisors on Varieties | 255 |
5 Néron Models | 318 |
6 Existence of Néron Models | 325 |
7 Intersection Theory Minimal Models and BlowingUp | 338 |
8 The Special Fiber of a Néron Model | 350 |
9 Tates Algorithm to Compute the Special Fiber | 361 |
10 The Conductor of an Elliptic Curve | 379 |
11 Oggs Formula | 389 |
Exercises | 396 |
CHAPTER V | 408 |
3 The Tate Curve | 422 |
4 The Tate Map Is Surjective | 430 |
5 Elliptic Curves over padic Fields | 438 |
6 Some Applications of padic Uniformization | 445 |
Exercises | 448 |
CHAPTER VI | 454 |
2 Local Decomposition of the Canonical Height | 461 |
4 NonArchimedean Absolute Values Explicit Formulas | 469 |
APPENDIX | 481 |
| 488 | |
List of Notation | 498 |
| 504 | |
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Common terms and phrases
abelian extension algebraic analytic arithmetic surface canonical height CL(RK coefficients completes the proof complex multiplication components compute curve with complex cusp form Dedekind definition discrete valuation ring Div(S divisor E₁ elliptic curve elliptic curve defined elliptic surface End(E endomorphism Eo(K example exercise form of weight formula Fourier fractional ideal function field Gal(K/K gives Grössencharacter group scheme Hartshorne Hecke operator height function Hence homomorphism idele intersection irreducible isogeny isomorphism j-invariant L-series lattice Lemma let E/K meromorphic minimal modular form modular function modulo morphism Néron model non-singular non-zero notation number field p-adic polynomial prime PROOF of Theorem Proposition Prove quadratic imaginary field R-scheme rational map ray class field reduction residue field ring of integers satisfies says special fiber subgroup Te(E unramified Weierstrass equation weight 2k