### 博碩士論文 91241001 詳細資訊

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(The Average Number of Torsion Points on Elliptic Curves)

 ★ 數論在密碼學上的應用 ★ a^n-b^n的原質因子，其中a,b為高斯整數 ★ Group Representations on GL(2,F_q) ★ Legendre的定理在Z[i]和Z[w]的情形 ★ Diophantine approximation and the Markoff chain ★ The average of the number of r-periodic points over a quadratic number field. ★ 週期為r之週期點個數的平均值 ★ 正特徵值函數體上的逼近指數之研究 ★ On some problem in Arithmetic Dynamical System and Diophantine Approximation in Positive Characteristic ★ ZCm 的理想環生成元個數之上限

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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。

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.

★ 橢圓曲線
★ 複乘

★ Elliptic Curve
★ Torsion Point

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

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