Elliptic curves : function theory, geometry, arithmetic

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Elliptic curves : function theory, geometry, arithmetic / Henry McKean, Victor Moll

資料種別:

図書

出版情報:

New York : Cambridge University Press, 1997

形態:

xiii, 280 p. ; 24 cm

著者名:

ISBN:

9780521582285 [0521582288] (: hd)
9780521658171 [0521658179] (: pb)

書誌ID:

BA31099447

子書誌情報

フルテキスト

所蔵情報

他の版・巻

書誌詳細

注記:

Includes bibliographical references (p. 265-277) and index

主題:

Curves, Elliptic

言語:

英語

目次情報:

Preface

First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves / 1.：

The Riemann Sphere / 1.1：

Complex Manifolds / 1.2：

Rational Functions / 1.3：

Luroth's Theorem / 1.4：

Automorphisms of P[superscript 1] / 1.5：

Spherical Geometry / 1.6：

Finite Subgroups and the Platonic Solids / 1.7：

Automorphisms of the Half-Plane / 1.8：

Hyperbolic Geometry / 1.9：

Projective Curves / 1.10：

Covering Surfaces / 1.11：

Scissors and Paste / 1.12：

Algebraic Functions / 1.13：

Examples / 1.14：

More on Uniformization / 1.15：

Compact Manifolds as Curves: Finale / 1.16：

Elliptic Integrals and Functions / 2.：

Elliptic Integrals: Where They Come From / 2.1：

The Incomplete Integrals Reduced to Normal Form / 2.2：

The Complete Integrals: Landen, Gauss, and the Arithmetic-Geometric Mean / 2.3：

The Complete Elliptic Integrals: Legendre's Relation / 2.4：

The Discovery of Gauss and Abel / 2.5：

Periods in General / 2.6：

Elliptic Functions in General / 2.7：

The and-Function / 2.8：

Elliptic Integrals, Complete and Incomplete / 2.9：

Two Mechanical Applications / 2.10：

The Projective Cubic / 2.11：

The Problem of Inversion / 2.12：

The Function Field / 2.13：

Addition on the Cubic / 2.14：

Abel's Theorem / 2.15：

Jacobian Functions: Reprise / 2.16：

Covering Tori / 2.17：

Finale: Higher Genus / 2.18：

Theta Functions / 3.：

Jacobi's Theta Functions / 3.1：

Some Identities / 3.2：

The Jacobi and Weierstrass Connections / 3.3：

Projective Embedding of Tori / 3.4：

Products / 3.5：

Sums of Two Squares / 3.6：

Sums of Four Squares / 3.7：

Euler's Identities: Partitio Numerorum / 3.8：

Jacobi's and Higher Substitutions / 3.9：

Quadratic Reciprocity / 3.10：

Ramanujan's Continued Fractions / 3.11：

Modular Groups and Modular Functions / 4.：

The Modular Group of First Level / 4.1：

The Modular Group of Second Level / 4.2：

Fundamental Cells / 4.3：

Generating the Groups / 4.4：

Gauss on Quadratic Forms / 4.5：

The Group of Anharmonic Ratios / 4.6：

Modular Forms / 4.7：

Eisenstein Sums / 4.8：

Absolute Invariants / 4.9：

Triangle Functions / 4.10：

The Modular Equation of Level 2 / 4.11：

Landen's Transformation / 4.12：

Modular Equations of Higher Level / 4.13：

Jacobi's Modular Equation / 4.14：

Jacobi and Legendre's Derivation: Level 5 / 4.15：

Arithmetic Subgroups: Overview / 4.16：

Ikosaeder and the Quintic / 5.：

Solvability of Equations of Degree [less than] 4 / 5.1：

Galois Groups Revisited / 5.2：

The Galois Group of Level 5 / 5.3：

An Element of Degree 5 / 5.4：

Hermite on the Depressed Equation / 5.5：

Hermite on the Quintic / 5.6：

A Geometric View / 5.7：

Imaginary Quadratic Number Fields / 6.：

Algebraic Numbers / 6.1：

Primes and Ideal Numbers / 6.2：

Class Invariants and Kronecker's Jugendtraum / 6.3：

Application of the Modular Equation / 6.4：

The Class Polynomial / 6.5：

Class Invariants at a Prime Level / 6.6：

Irreducibility of the Class Polynomial / 6.7：

Class Field and Galois Group / 6.8：

Computation of the Class Invariants / 6.9：

Arithmetic of Elliptic Curves / 7：

Arithmetic of the Projective Line / 7.1：

Cubics: The Mordell--Weil Theorem / 7.2：

Proof of the Mordell--Weil Theorem / 7.3：

References

Index

Preface

First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves / 1.：

The Riemann Sphere / 1.1：

Complex Manifolds / 1.2：

Rational Functions / 1.3：

Luroth's Theorem / 1.4：

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