計算橢圓曲線上的 Mordell-Weil 群

On the computation of the Mordell-Weil groups of elliptic curves

蔡東霖(Tung-Lin Tsai)

指導教授 ： 紀文鎮

國立臺灣師範大學/理學院/數學系/碩士(2003年)

摘要

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。

關鍵字

橢圓曲線 ； 基底 ； HASH(0xdd71448) ； HASH(0xdd714c0) ； HASH(0xdd71538) ； HASH(0xdd715b0) ； HASH(0xdd71f0c) ； HASH(0xdd71f84)

並列摘要

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.

並列關鍵字

Mordell-Weil ； Manin's theorem ； elliptic curve ； Selmer group ； Shafarevich-Tate group ； LLL-reduction ； Galois cohomology ； Basis of elliptic curves

參考文獻

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[GZ] Josef Gebel and Horst G. Zimmer Computing the Mordell-Weil group of an elliptic curve over , CRM

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[Gr] D. R. Grayson, The arithogeometric mean, Arch. Math. 52 (1989), 507-512.