Elliptic CurvesSpringer Science & Business Media, 6 ביוני 2006 - 490 עמודים There are three new appendices, one by Stefan Theisen on the role of Calabi– Yau manifolds in string theory and one by Otto Forster on the use of elliptic curves in computing theory and coding theory. In the third appendix we discuss the role of elliptic curves in homotopy theory. In these three introductions the reader can get a clue to the far-reaching implications of the theory of elliptic curves in mathematical sciences. During the ?nal production of this edition, the ICM 2002 manuscript of Mike Hopkins became available. This report outlines the role of elliptic curves in ho- topy theory. Elliptic curves appear in the form of the Weierstasse equation and its related changes of variable. The equations and the changes of variable are coded in an algebraic structure called a Hopf algebroid, and this Hopf algebroid is related to a cohomology theory called topological modular forms. Hopkins and his coworkers have used this theory in several directions, one being the explanation of elements in stable homotopy up to degree 60. In the third appendix we explain how what we described in Chapter 3 leads to the Weierstrass Hopf algebroid making a link with Hopkins’ paper. |
תוכן
| 1 | |
Elementary Properties of the ChordTangent Group | 23 |
Plane Algebraic Curves 45 | 44 |
Factorial Rings and Elimination Theory | 57 |
Elliptic Curves and Their Isomorphisms | 65 |
Families of Elliptic Curves and Geometric Properties | 85 |
Reduction mod p and Torsion Points | 103 |
Proof of Mordells Finite Generation Theorem 125 | 124 |
Faltings Finiteness Theorem | 303 |
Tates Conjecture ˇSafareviˇcs Theorem and Faltings Proof | 305 |
Serres Open Image Theorem | 307 |
LFunction of an Elliptic Curve and Its Analytic Continuation | 309 |
Zeta Functions of Curves over Q | 310 |
HasseWeil LFunction and the Functional Equation | 312 |
Classical Abelian LFunctions and Their Functional Equations | 315 |
Grossencharacters and Hecke LFunctions | 318 |
Galois Cohomology and Isomorphism Classification | 143 |
Descent and Galois Cohomology | 157 |
Elliptic and Hypergeometric Functions | 167 |
Modular Functions 209 | 208 |
The Modular Curves XN X1N and X0N | 215 |
The LFunction of a Modular Form | 222 |
New Forms | 229 |
Endomorphisms of Elliptic Curves | 233 |
Symplectic Pairings on Lattices and Division Points | 235 |
Isogenies in the General Case | 237 |
Endomorphisms and Complex Multiplication | 241 |
The Tate Module of an Elliptic Curve | 245 |
Endomorphisms and the Tate Module | 246 |
Expansions Near the Origin and the Formal Group | 248 |
Elliptic Curves over Finite Fields | 253 |
Generalities on Zeta Functions of Curves over a Finite Field | 256 |
Definition of Supersingular Elliptic Curves | 259 |
Number of Supersingular Elliptic Curves | 263 |
Points of Order p and Supersingular Curves | 265 |
The Endomorphism Algebra and Supersingular Curves | 266 |
Summary of Criteria for a Curve To Be Supersingular | 268 |
Tates Description of Homomorphisms | 270 |
Division Polynomial | 272 |
Elliptic Curves over Local Fields 275 | 274 |
The Neron Minimal Model | 277 |
Galois Criterion of Good Reduction of NeronOggˇSafareviˇc | 281 |
Elliptic Curves over the Real Numbers | 284 |
Elliptic Curves over Global Fields and lAdic Representations | 291 |
Generalities on lAdic Representations | 293 |
Galois Representations and the NeronOggˇSafareviˇc Criterion in the Global Case | 296 |
ˇCebotarevs Density Theorem | 298 |
Variation of l | 301 |
Deurings Theorem on the LFunction of an Elliptic Curve with Complex Multiplication | 321 |
EichlerShimura Theory | 322 |
The Modular Curve Conjecture | 324 |
Remarks on the Birch and SwinnertonDyer Conjecture | 325 |
Rank Conjecture for Curves with Complex Multiplication I by Coates and Wiles | 326 |
Rank Conjecture for Curves with Complex Multiplication II by Greenberg and Rohrlich | 327 |
Rank Conjecture for Modular Curves by Gross and Zagier | 328 |
The Conjecture of Birch and SwinnertonDyer on the Leading Term | 329 |
Heegner Points and the Derivative of the Lfunction at s 1 after Gross and Zagier | 330 |
October 1986 | 331 |
Remarks on the Modular Elliptic Curves Conjecture and Fermats Last Theorem | 333 |
Semistable Curves and Tate Modules | 334 |
The Frey Curve and the Reduction of Fermat Equation to Modular Elliptic Curves over Q | 335 |
Modular Elliptic Curves and the Hecke Algebra | 336 |
Hecke Algebras and Tate Modules of Modular Elliptic Curves | 338 |
Special Properties of mod 3 Representations | 339 |
Properties of the Universal Deformation Ring | 341 |
Remarks on the Proof of the Opposite Inequality | 342 |
CalabiYau Varieties 345 | 344 |
Real Differential Geometry | 347 |
Complex Differential Geometry | 349 |
Kahler Manifolds | 352 |
Connections Curvature and Holonomy | 356 |
Projective Spaces Characteristic Classes and Curvature | 361 |
Families of Elliptic Curves | 383 |
CalabiYau Manifolds and String Theory 403 | 402 |
Elliptic Curves in Algorithmic Number Theory | 413 |
Elliptic Curves and Topological Modular Forms | 425 |
Guide to the Exercises | 445 |
References | 465 |
List of Notation | 479 |
מהדורות אחרות - הצג הכל
מונחים וביטויים נפוצים
abelian assertion automorphism bad reduction Calabi–Yau manifolds calculate change of variable Chapter characteristic Chern class coefficients cohomology commutative complex multiplication complex numbers conic conjecture consider corresponding cubic curve cubic equation Definition degree denote divisor elements elliptic curve defined elliptic function End(E equivalent Euler product exact sequence example extension factor fibration fibre finite field formal group formula Gal(ks/k given group law Hecke Hence homomorphism Hopf algebroid hypersurface integer intersection points irreducible isogeny isomorphism j-invariant K¨ahler K3 surfaces l-adic representation L-function line bundle minimal modular curve modular forms modulo morphism nonsingular nonzero normal form notation number field ordp(x polynomial prime projective space Proof quadratic quotient rational numbers rational points relation Remark ring roots satisfies singular smooth string theory structure subgroup supersingular Tate module torsion unramified vector Weierstrass Weierstrass equation zero zeta function