橢圓曲線算術

內容概要

　　美國哈佛大學從1977年以來曾多次舉辦“橢圓曲線”班，《橢圓曲線算術(第2版)(英文版)》作者是該討論班成員之一。橢圓曲線是一個古老的數(shù)學課題，最近由于代數(shù)數(shù)論和代數(shù)幾何等現(xiàn)代數(shù)學的進展，使它得到了新的活力。本書則是以上述觀點處理橢圓函數(shù)的算術理論，包括橢圓曲線的幾何背景，橢圓曲線的形式群，有限域上的橢圓函數(shù)、復數(shù)、局部域和整體域等基本內容，最后兩章討論整數(shù)和有理數(shù)。書末有三個附錄。這是第二版，在第一版的基礎上增加了“橢圓曲線的代數(shù)方面“全新一章，重在強調有限域上的算術，包括lenstra因式分解算術，schoof點計算算術，計算tate和weil派對的miller算術。新增加了一部分講述szpiró猜想和abc，擴展和更新了大量的最新進展和大量新的練習。目次：代數(shù)變量；代數(shù)曲線；橢圓曲線幾何；橢圓曲線的標準群；有限域上的橢圓曲線；c上的橢圓曲線；局部域上的橢圓曲線；全局域上的橢圓曲線；橢圓曲線的整數(shù)點；mordell-weil群上的計算；橢圓曲線的算術方面。
　　讀者對象：數(shù)學專業(yè)的研究生教材、科研人員和相關的科技工作者。

作者簡介

作者：(美國)希爾弗曼 (Joseph H.Silverman)

書籍目錄

preface to the second edition
preface to the first edition
introduction
chapter i algebraic varieties
§1. affine varieties
§2. projective varieties
§3. maps between varieties
exercises
chapter ii algebraic curves
§1. curves
§2. maps between curves
§3. divisors
§4. differentials
§5. the riemann-roch theorem
exercises
chapter iii the geometry of elliptic curves
§1. weierstrass equations
§2. the group law
§3. elliptic curves
.§4. isogenies
§5. the invariant differential
§6. the dual isogeny
§7. the tate module
§8. the weil pairing
§9. the endomorphism ring
§ 10. the automorphism group
exercises
chapter iv the formal group of an elliptic curve
§ 1. expansion around o
§2. formal groups
§3. groups associated to formal groups
§4. the invariantdifferential
§5. the formal logarithm
§6. formal groups over discrete valuation rings
§7. formal groups in characteristic p
exercises
chapter v elliptic curves over finite fields
§ 1. number of rational points
§2. the weil conjectures
§3. the endomorphism ring
§4. calculating the hasse invariant
exercises
chapter vi elliptic curves over c
§1. elliptic integrals
§2. elliptic functions
§3. construction of elliptic functions
§4. maps analytic and maps algebraic
§5. uniformization
§6. the lefschetz principle
exercises
chapter vii elliptic curves over local fields
§1. minimal weierstrass equatlons
§2. reduction modulo
§3. points of finite order
§4. the action of inertia
§5. good and bad reduction
§6. the croup e/e0
§7. the criterion of n~ron-ogg-shafarevich
exercises
chapter viii elliptic curves over global fields
§1. the weak mordell-weil theorem
§2. the kummer pairing via cohomology
§3. the descent procedure
§4. the mordell-weil theorem over q
§5. heights on projective space
§6. heights on elliptic curves
§7. torsion points
§8. the minimal discriminant
§9. the canonical height
§10. the rank of an elliptic curve
§11. szpiro's conjecture and abc
exercises
chapter ix integral points on elliptic curves
§1. diophantine approximation
§2. distance functions
§3. siegel's theorem
§4. the s-unit equation
§5. effective methods
§6. shafarevich's theorem
§7. the curve ye = x3 + d
§8. roth's theorem--an overview
exercises
chapter x computing the mordell-weil group
§1. an example
§2. twisting--general theory
§3. homogeneous spaces
§4. the selmer and shafarevich-tate groups
§5. twisting--elliptic curves
§6. the curve y2 = xa + dx
exercises
chapter xi algorithmic aspects of elliptic curves
§1. double-and-add algorithms
§2. lenstra's elliptic curve factorization algorithm
§3. counting the number of points in e(fq)
§4. elliptic curve cryptography
§5. solving the ecdlp: the general case
§6. solving the ecdlp: special cases
§7. pairing-based cryptography
§8. computing the weil pairing
§9. the tatae-lichtenbanm pairing
exercises
appendix a elliptic curves in characteristics 2 and 3
exercises
appendix b group cohomology (ho and h1)
§1. cohomology of finite groups
§2. galois cohomology
§3. nonabelian cohomology
exercises
appendix c further topics: an overview
§11. complex multiplication
§12. modular functions
§13, modular curves
§14. tate curves
§15. n6ron models and tate's algorithm
§16. l-series
§17. duality theory
§18. local height functions
§19. the image of galois
§20. function fields and specialization theorems
§21. variation of ap and the sato-tate conjecture
notes on exercises
list of notation
references
index

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