橢圓曲線

內(nèi)容概要

　　橢圓曲線是映射解形成群的兩變量三次方程。模型形式是具有特定變換規(guī)律和增長(zhǎng)性質(zhì)的上半平面上的解析函數(shù)。橢圓曲線和模型形式兩大論題共同形成eichler-shimura理論，構(gòu)成了橢圓曲線特殊種類的模型性質(zhì)。該命題的逆命題——taniyama-weil猜想及其暗含的費(fèi)馬大定理，所有的有理橢圓曲線都來(lái)源于此。
　　目次：概論；射影空間中曲線；weierstrass形式中的立方曲線；mordell定理；e（q）的torsion子群；復(fù)點(diǎn)；dirichlet定理；sl(2,z)的模型定理；hecke子群的模型形式；橢圓曲線的l曲線；eichler-shimura理論；taniyama-weil猜想。
　　讀者對(duì)象：數(shù)學(xué)專業(yè)的高年級(jí)本科生、研究生和相關(guān)專業(yè)的科研人員。

作者簡(jiǎn)介

作者:(美)奈普

書籍目錄

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
xl. 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

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