Anthony W.Knapp所著的《椭圆曲线》阐述了椭圆曲线是映射解形成群的两变量三次方程。模型形式是具有特定变换规律和增长性质的上半平面上的解析函数。椭圆曲线和模型形式两大论题共同形成Eichler-Shimura理论，构成了椭圆曲线特殊种类的模型性质。该命题的逆命题——Taniyama-Weil猜想及其暗含的费马大定理，所有的有理椭圆曲线都来源于此。

list of figures

list of tables

preface

standard notation

i. overview

ii. curves in projective space

 1. projective space

 2. curves and tangents

 3. flexes

 4. application to cubics

 5. bezout's theorem and resultants

iii. cubic curves in weierstrass form

 1. examples

 2. weierstrass form, discriminant, j-invariant

 3. group law

 4. computations with the group law

 5. singular points

iv. mordell's theorem

 1. descent

 2. condition for divisibility by 2

 3. e(q)/2e(q), special case

 4. e(q)/2e(q), general case

 5. height and mordell's theorem

 6. geometric formula for rank

 7. upper bound on the rank

 8. construction of points in e(q)

 9. appendix on algebraic number theory

v. torsion subgroup of e(q)

 1. overview

 2. reduction modulo p

 3. p-adic filtration

 4. lutz-nagell theorem

 5. construction of curves with prescribed torsion

 6. torsion groups for special curves

vi. complex points

 1. overview

 2. elliptic functions

 3. weierstrass p function

 4. effect on addition

 5. overview of inversion problem

 6. analytic continuation

 7. riemann surface of the integrand

 8. an elliptic integral

 9. computability of the correspondence

vii. dirichlet's theorem

 1. motivation

 2. dirichlet series and euler products

 3. fourier analysis on finite abelian groups

 4. proof of dirichlet's theorem

 5. analytic properties of dirichlet l functions

viii. modular forms for sl(2, z)

 1. overview

 2. definitions and examples

 3. geometry of the q expansion

 4. dimensions of spaces of modular forms

 5. l function of a cusp form

 6. petersson inner product

 7. hecke operators

 8. interaction with petersson inner product

ix. modular forms for hecke subgroups

 1. hecke subgroups

 2. modular and cusp forms

 3. examples of modular forms

 4. l function of a cusp form

 5. dimensions of spaces of cusp forms

 6. hecke operators

 7. oldforms and newforms

 x. l function of an elliptic curve

 1. global minimal weierstrass equations

 2. zeta functions and l functions

 3. hasse's theorem

xi. eichler-shimura theory

 1. overview

 2. riemann surface xo(n)

 3. meromorphic differentials

 4. properties of compact riemann surfaces

 5. hecke operators on integral homology

 6. modular function j(r)

 7. varieties and curves

 8. canonical model of xo(n)

 9. abstract elliptic curves and isogenies

 10. abelian varieties and jacobian variety

 11. elliptic curves constructed from s2(γ0(n))

 12. match of l functions

xii. taniyama-weil conjecture

 1. relationships among conjectures

 2. strong wei] curves and twists

 3. computations of equations of well curves

 4. connection with fermat's last theorem

notes

references

index of notation

index