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    Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/43900


    Title: 橢圓曲線上扭點的平均數;The Average Number of Torsion Points on Elliptic Curves
    Authors: 黃秀戀;Hsiu-Lien Huang
    Contributors: 數學研究所
    Keywords: 扭點;橢圓曲線;複乘;Complex Multiplication;Elliptic Curve;Torsion Point
    Date: 2010-06-23
    Issue Date: 2010-12-08 14:25:56 (UTC+8)
    Publisher: 國立中央大學
    Abstract: 令 E 是橢圓曲線定義在數體 K 上, E[m](Fp) 是橢圓曲線 E 在有限體 Fp上的m-扭點。本篇論文中,我們想要計算橢圓曲線扭點個數的平均數:即 E[m](Fp) 元素個數對所有質數 p 的平均值。 在第二章,我們簡單的介紹橢圓曲線,及有複乘 (complex multiplication) 的橢圓曲線所需要用的代數數論知識。 我們分別計算了有複乘和沒有複乘之橢圓曲線的情況。 在第三四章,就有複乘的橢圓曲線,我們研究了兩類橢圓曲 Y²=X³-DX 和 Y²=X³+A,仔細計算它們的伽羅瓦群 (Galois group)。就第一類的橢圓曲線,我們能夠完全的決定它的伽羅瓦群。第二類的橢圓曲線,我們能夠決定絕大部分的伽羅瓦群。再利用伽羅瓦群的個數,去計算我們原來想算的平均數。 在第五章,我們利用伽羅瓦群的群作用,計算在有複乘和沒有複乘的情況下,求出我們想計算的平均數。 Given an elliptic curve E defined over a number field K and an integer m, let E[m] be the m-torsion subgroup of E. For a prime ideal p in K, let Fp be the residue field of R at p, where R be the ring of integer of K. Let E[m](Fp) be the set of m-torsion points of E that are rational over Fp, and #E[m](Fp) denote the number element in E[m](Fp). Our main goal is to compute the ratio of the total sum of #E[m](Fp) and the number of all prime ideal ins in K which does not divide m and the discriminant of E. We focus on two families of elliptic curves Y²=X³-DX and Y²=X³+A, D,A are nonzero interger, which are families of elliptic curves with complex multiplication by the ring of Gaussian integers or the ring of Eisenstein integers respectively. One of the major reasons for us to focus on the above two families is because there are explicit formulas of Grossencharacter which are attached to these two families.
    Appears in Collections:[數學研究所] 博碩士論文

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