椭圆曲线算术

preface to the second editionpreface to the first editionintroductionchapter i algebraic varieties§1. affine varieties§2. projective varieties§3. maps between varietiesexerciseschapter ii algebraic curves§1. curves§2. maps between curves§3. divisors§4. differentials§5. the riemann-roch theoremexerciseschapter 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 groupexerciseschapter 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 pexerciseschapter v elliptic curves over finite fields§ 1. number of rational points§2. the weil conjectures§3. the endomorphism ring§4. calculating the hasse invariantexerciseschapter 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 principleexerciseschapter 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-shafarevichexerciseschapter 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 abcexerciseschapter 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 overviewexerciseschapter 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 dxexerciseschapter 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 pairingexercisesappendix a elliptic curves in characteristics 2 and 3exercisesappendix b group cohomology (ho and h1)§1. cohomology of finite groups§2. galois cohomology§3. nonabelian cohomologyexercisesappendix 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 conjecturenotes on exerciseslist of notationreferencesindex

　　美国哈佛大学从1977年以来曾多次举办“椭圆曲线”班，《椭圆曲线算术(第2版)(英文版)》作者是该讨论班成员之一。椭圆曲线是一个古老的数学课题，最近由于代数数论和代数几何等现代数学的进展，使它得到了新的活力。本书则是以上述观点处理椭圆函数的算术理论，包括椭圆曲线的几何背景，椭圆曲线的形式群，有限域上的椭圆函数、复数、局部域和整体域等基本内容，最后两章讨论整数和有理数。书末有三个附录。这是第二版，在第一版的基础上增加了“椭圆曲线的代数方面“全新一章，重在强调有限域上的算术，包括lenstra因式分解算术，schoof点计算算术，计算tate和weil派对的miller算术。新增加了一部分讲述szpiró猜想和abc，扩展和更新了大量的最新进展和大量新的练习。目次：代数变量；代数曲线；椭圆曲线几何；椭圆曲线的标准群；有限域上的椭圆曲线；c上的椭圆曲线；局部域上的椭圆曲线；全局域上的椭圆曲线；椭圆曲线的整数点；mordell-weil群上的计算；椭圆曲线的算术方面。　　读者对象：数学专业的研究生教材、科研人员和相关的科技工作者。