# User:Z423x5c6/橢圓曲線

${\displaystyle y^{2}=x^{3}+ax+b}$

## 實數上嘅橢圓曲線

${\displaystyle y^{2}=x^{3}+ax+b}$

${\displaystyle \Delta =-16(4a^{3}+27b^{2})}$

## 羣運算

${\displaystyle s={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}}$

${\displaystyle (sx+d)^{2}=x^{3}+ax+b}$

${\displaystyle (x-x_{P})(x-x_{Q})(x-x_{R})=x^{3}+x^{2}(-x_{P}-x_{Q}-x_{R})+x(x_{P}x_{Q}+x_{P}x_{R}+x_{Q}x_{R})-x_{P}x_{Q}x_{R}}$

{\displaystyle {\begin{aligned}x_{R}&=s^{2}-x_{P}-x_{Q}\\y_{R}&=y_{P}+s(x_{R}-x_{P})\end{aligned}}}

{\displaystyle {\begin{aligned}s&={\frac {3{x_{P}}^{2}+a}{2y_{P}}}\\x_{R}&=s^{2}-2x_{P}\\y_{R}&=y_{P}+s(x_{R}-x_{P})\end{aligned}}}

## 複數上面嘅橢圓曲線

${\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}$

${\displaystyle z\mapsto [1:\wp (z):\wp '(z)/2]}$

${\displaystyle y^{2}=x(x-1)(x-\lambda )}$

${\displaystyle g_{2}={\frac {\sqrt[{3}]{4}}{3}}(\lambda ^{2}-\lambda +1)}$

${\displaystyle g_{3}={\frac {1}{27}}(\lambda +1)(2\lambda ^{2}-5\lambda +2)}$

${\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}=\lambda ^{2}(\lambda -1)^{2}}$

${\displaystyle {\frac {a}{n}}\omega _{1}+{\frac {b}{n}}\omega _{2}}$

## 有理數上面嘅橢圓曲線

### 有理數點羣嘅結構

${\displaystyle E(\mathbb {Q} )}$嘅階（rank），即係${\displaystyle E(\mathbb {Q} )}$入面有幾多個${\displaystyle \mathbb {Z} }$嘅因子，又或者另一個講法，獨立嘅無限階元素嘅數量，有時亦都會簡單叫做E嘅階。千禧年大獎難題其中之一嘅Birch同Swinnerton-Dyer猜想就係同計算階有關嘅。雖然已知嘅橢圓曲線嘅階都係好細，但係佢地猜想其實個階係幾大都有可能。現時確切知道階最大嘅橢圓曲線係

y2 + xy + y = x3x2244537673336319601463803487168961769270757573821859853707x + 961710182053183034546222979258806817743270682028964434238957830989898438151121499931

### Birch同Swinnerton-Dyer猜想

Brich同Swinnerton-Dyer猜想係克雷數學研究所七條千禧年難題之一，呢個猜想同時牽涉到由橢圓曲線整出嚟嘅分析上同算術上嘅物體。

${\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}$

${\displaystyle Z(E(\mathbf {F} _{p}))=\exp \left(\sum \#\left[E({\mathbf {F} }_{p^{n}})\right]{\frac {T^{n}}{n}}\right)}$

${\displaystyle Z(E(\mathbf {F} _{p}))={\frac {1-a_{p}T+pT^{2}}{(1-T)(1-pT)}},}$

${\displaystyle a_{p}=p+1-\#E(\mathbb {F} _{p}).}$

E係${\displaystyle \mathbb {Q} }$上嘅Hasse-Weil ζ函數就係將所有質數p嘅呢個資訊收集埋一齊，佢嘅定義係

${\displaystyle L(E(\mathbf {Q} ),s)=\prod _{p}\left(1-a_{p}p^{-s}+\varepsilon (p)p^{1-2s}\right)^{-1}}$
##### 其中如果E喺p係「好商餘」嘅話${\displaystyle \varepsilon (p)=1}$，壞就係0，嗰陣${\displaystyle a_{p}}$要用另一個方法定義，可以睇Silverman (1986)。

• 一個全等數（congruent number）嘅定義係佢可以寫做一個邊長係有理數嘅直角三角形嘅面積，而一個已知嘅結果係，n係一個全等數若且唯若橢圓曲線${\displaystyle y^{2}=x^{3}-n^{2}x}$有一個無限階嘅有理點；如果BSD係啱嘅話，噉就等價於佢嘅L-函數喺s=1嗰點係0。Tunnell證明咗一個相關嘅定理：如果BSD係啱嘅話，n係一個全等數若且唯若符合${\displaystyle 2x^{2}+y^{2}+8z^{2}=n}$嘅點嘅數量係符合${\displaystyle 2x^{2}+y^{2}+32z^{2}=n}$嘅兩倍。呢個定理嘅有用之處在於呢個條件好容易就檢查到[10]
• 喺另一個方向，用一啲分析上嘅方法可以估算到某啲L-函數喺關鍵帶（critical strip）中間嘅零點嘅階，假設BSD嘅話，呢啲估算對應住一啲橢圓曲線族嘅階嘅資訊。例如，假設廣義黎曼猜想同BSD係啱嘅話，${\displaystyle y^{2}=x^{3}+ax+b}$${\displaystyle y^{2}=x^{3}+ax+b}$${\displaystyle y^{2}=x^{3}+ax+b}$${\displaystyle y^{2}=x^{3}+ax+b}$呢一族嘅橢圓曲線嘅平均階細過2[11]

### 模定理同埋喺費馬大定理上面嘅應用

${\displaystyle L(E(\mathbf {Q} ),s)=\sum _{n>0}a(n)n^{-s},}$

${\displaystyle \sum a(n)q^{n},\qquad q=e^{2\pi iz}}$

## 拎

[[Category:羣論]] [[Category:解析數論]]

1. Wing Tat Chow, Rudolf (2018). "The Arithmetic-Geometric Mean and Periods of Curves of Genus 1 and 2" (PDF). White Rose eTheses Online. p. 12.
2. Silverman 1986, Theorem 4.1
3. Silverman 1986, pp. 199–205
4. See also J. W. S. Cassels, Mordell's Finite Basis Theorem Revisited, Mathematical Proceedings of the Cambridge Philosophical Society 100, 3–41 and the comment of A. Weil on the genesis of his work: A. Weil, Collected Papers, vol. 1, 520–521.
5. Dujella, Andrej. "History of elliptic curves rank records". University of Zagreb.
6. Silverman 1986, Theorem 7.5
7. Silverman 1986, Remark 7.8 in Ch. VIII
8. The definition is formal, the exponential of this power series without constant term denotes the usual development.
9. see for example Silverman, Joseph H. (2006). "An Introduction to the Theory of Elliptic Curves" (PDF). Summer School on Computational Number Theory and Applications to Cryptography. University of Wyoming.
10. Koblitz 1993
11. Heath-Brown, D. R. (2004). "The Average Analytic Rank of Elliptic Curves". Duke Mathematical Journal. 122 (3): 591–623. arXiv:math/0305114. doi:10.1215/S0012-7094-04-12235-3.