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研究生： 蔡東霖
Tsai, Tung-Lin
論文名稱： 計算橢圓曲線上的 Mordell-Weil 群
On the computation of the Mordell-Weil groups of elliptic curves
指導教授： 紀文鎮
Chi, Wen-Chen
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2004
畢業學年度： 92
語文別： 中文
論文頁數： 46
中文關鍵詞： 橢圓曲線 、基底
英文關鍵詞： Mordell-Weil, Manin's theorem, elliptic curve, Selmer group, Shafarevich-Tate group, LLL-reduction, Galois cohomology, Basis of elliptic curves
論文種類： 學術論文
相關次數： 點閱：143  下載：12 
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    • 內容主要分成三部分，第一部分介紹一些後面章節會用到的基本定義以及定理。第二部分介紹如何透過計算 Selmer 群跟 Shafarevich-Tate 群來計算 Mordell-Weil 群的一組基底。第三部分則是介紹如何透過 Manin's 算則來計算Mordell-Weil 群的一組基底。

    From some basic results of Algebraic Number Theory and Algebraic Geometry, we know
    that the set of points of an elliptic curve E over Q has a group structure, and the rational
    points form a subgroup (called the Mordell-Weil group) which is denoted by E(Q). Moreover,
    by the well-known Mordell-Weil Theorem, we know that E(Q) is finitely generated. It arises
    a question that can we find a basis for the Mordell-Weil group? Unfortunately, for the time
    being, there is no any valid algorithm for computing a basis for the Mordell-Weil group.
    However, several methods to find a basis for Mordell-Weil groups are available under some
    specific conditions. In this thesis, we would like to study two of them.
    The content is divided into three sections. In section one, we recall some basic definitions
    and theorems that would be used in the remaining sections, including basic properties of
    elliptic curves, height functions, and group cohomology. In section two, we study how to
    compute a basis of Mordell-Weil group via phi-Selmer group and Shafarevich-Tate group under
    the hypothesis that f(x) has rational roots, where the elliptic curve E is given by y2 = f(x).
    We would discuss the case where f(x) has only one rational root and the case where f(x)
    has three rational roots. In each case, we shall give an example. Finally, in section three, we
    would study computing a basis of Mordell-Weil group via Manin’s conditional algorithm.

    1. Preliminaries 3 1.1. Elliptic curves 3 1.2. Heights 4 1.3. Group cohomology 6 2. Computing Mordell-Weil groups via -Selmer groups and Shafarevich-Tategroups. 10 2.1. There is only one rational root for f(x) 12 2.2. There are three rational roots for f(x) 29 3. Computing Mordell-Weil groups via Manin’s conditional algorithm. 36 3.1. Estimating the upper bounds for the rank r and the regulator R 36 3.2. Manin’s theorem 40 3.3. Finding a generating set of ˆE(Q) 42 3.4. Finding a basis of E(Q) 43 References 46

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