橢圓曲線上扭點的平均數

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黃秀戀(Hsiu-Lien Huang) 查詢紙本館藏

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數學系

論文名稱

橢圓曲線上扭點的平均數
(The Average Number of Torsion Points on Elliptic Curves)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 扭點
★ 橢圓曲線
★ 複乘

關鍵字(英)

★ Complex Multiplication
★ Elliptic Curve
★ Torsion Point

論文目次

1 Introduction.................................................................................................................1
2 Elliptic curve with complex multiplication.....................................................................6
2.1 Elliptic curves........................................................................................................6
2.2 Complex Multiplication..........................................................................................8
2.3 Class Field Theory................................................................................................10
2.3.1 Quartic reciprocityla.....................................................................................13
2.3.2 Cubic Residue Symbol.................................................................................14
2.3.3 Sextic reciprocity law...................................................................................16
2.3.4 The Idelic Formulation of Class Field Theory................................................19
2.4 Abelian Extension.................................................................................................20
2.5 The Main Theorem of Complex Multiplication.......................................................21
2.6 The Associate Grossencharacter.............................................................................22
3 The Average Number of Torsion Points on Elliptic Curves with CM by Q[i]..................26
3.1 The Average Number of Torsion Points..................................................................27
3.2 The Average Number of Torsion Points on Elliptic Curves Y²=X³-DX....................29
3.3 Fields of torsion points..........................................................................................35
3.4 The Average Number of Torsion Points over Qp in Special Elliptic Curve...............47
4 The Average Number of Torsion Points of Elliptic Curves with CM by Q[ω]................54
4.1 The Average Number of torsion points of Y²=X³+A..............................................55
4.2 Fields of torsion points of elliptic curve with CM by Q[ω]......................................72
5 Group Actions............................................................................................................95
5.1 Preliminary...........................................................................................................95
5.2 Galois actions.......................................................................................................96
5.2.1 Elliptic Curves without Complex Multiplication............................................97
5.2.2 Elliptic Curves with Complex Multiplication...............................................101

參考文獻

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指導教授

夏良忠(Liang-Chung Hsia)

審核日期

2010-6-23

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