Elliptic curve – github-backend

旧博客原文

原题：A crash introduction to BSD conjecture

The pdf version is A crash introduction to BSD conjecture .

We begin with the Weierstrass form of elliptic equation, i.e. look it as an embedding cubic curve in ${\mathop{\mathbb P}^2}$ .

Definition 1 (Weierstrass form) ${E \hookrightarrow \mathop{\mathbb P}^2 }$ , In general the form is given by,

$\displaystyle E: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6 \ \ \ \ \ (1)$

If ${char F \neq 2,3}$ , then, we have a much more simper form,

$\displaystyle y^2=x^3+ax+b, \Delta:=4a^3+27b^2\neq 0. \ \ \ \ \ (2)$

Remark 1

$\displaystyle \Delta(E)=\prod_{1\leq i,\neq j\leq 3}(z_i-z_j)$

Where ${z_i^3+az_i+b=0, \forall 1\leq i\leq 3}$ .

We have two way to classify the elliptic curve ${E}$ living in a fix field ${F}$ . \paragraph{j-invariant} The first one is by the isomorphism in ${\bar F}$ . i.e. we say two elliptic curves ${E_1,E_2}$ is equivalent iff

$\displaystyle \exists \rho:\bar F\rightarrow \bar F$

is a isomorphism such that ${\rho(E_1)=E_2}$ .

Definition 2 (j-invariant) For a elliptic curve ${E}$ , we have a j-invariant of ${E}$ , given by,

$\displaystyle j(E)=1728\frac{4a^3}{4a^3+27b^2} \ \ \ \ \ (3)$

Why j-invariant is important, because j-invariant is the invariant depend the equivalent class of ${E}$ under the classify of isomorphism induce by ${\bar F}$ . But in one equivalent class, there also exist a structure, called twist.

Definition 3 (Twist) For a elliptic curve ${E:y^2=x^3+ax+b}$ , all elliptic curve twist with ${E}$ is given by,

$\displaystyle E^{(d)}:y^2=x^3+ad^2x+bd^3 \ \ \ \ \ (4)$

So the twist of a given elliptic curve ${E}$ is given by:

$\displaystyle H^1(Gal(\bar F/ F), Aut(E_{\bar F})) \ \ \ \ \ (5)$

Remark 2 Of course a elliptic curve ${E:y^2=x^3+ax+b}$ is the same as ${E:y^2=x^3+ad^2x+bd^4}$ , induce by the map ${\mathop{\mathbb P}^1\rightarrow \mathop{\mathbb P}^1, (x,y,1)\rightarrow (x,dy,1)}$ .

But this moduli space induce by the isomorphism of ${F}$ is not good, morally speaking is because of the abandon of universal property. see \cite{zhang}. \paragraph{Level ${n}$ structure} We need a extension of the elliptic curve ${E}$ , this is given by the integral model.

Definition 4 (Integral model) ${s:=Spec(\mathcal{O}_F)}$ , ${E\rightarrow E_s}$ . ${E_s}$ is regular and minimal, the construction of ${E_s}$ is by the following way, we first construct ${\widetilde{E_s} }$ and then blow up. ${\widetilde E_s}$ is given by the Weierstrass equation with coefficent in ${\mathcal{O}_F}$ .

Remark 3 The existence of integral model need Zorn’s lemma.

Definition 5 (Semistable) the singularity of the minimal model of ${E}$ are ordinary double point.

Remark 4 Semistable is a crucial property, related to Szpiro’s conjecture.

Definition 6 (Level ${n}$ structure)

$\displaystyle \phi: ({\mathbb Z}/n{\mathbb Z})_s^2\longrightarrow E[N] \ \ \ \ \ (6)$

${P=\phi(1,0), Q=\phi(o,1)}$ The weil pairing of ${P,Q}$ is given by a unit in cycomotic fields, i.e. ${<P,Q>=\zeta_N\in \mu_{N}(s)}$

What happen if ${k={\mathbb C}}$ ? In this case we have a analytic isomorphism:

$\displaystyle E({\mathbb C})\simeq {\mathbb C}/\Lambda \ \ \ \ \ (7)$

Given by,

$\displaystyle {\mathbb C}/\Lambda \longrightarrow \mathop{\mathbb P}^2 \ \ \ \ \ (8)$

$\displaystyle z\longrightarrow (\mathfrak{P}(z), \mathfrak{P}'(z), 1 ) \ \ \ \ \ (9)$

Where ${\mathfrak{P(z)}=\frac{1}{z^2}+\sum_{\lambda\in \Lambda,\lambda\neq 0}(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2})}$ , and the Weierstrass equation ${E}$ is given by ${y^2=4x^3-60G_4(\Lambda)x-140G_6(\Lambda)}$ . The full n tructure of it is given by ${{\mathbb Z}+{\mathbb Z}\lambda}$ and the value of ${P,Q}$ , i.e.

$\displaystyle P=\frac{1}{N}, Q=\frac{\tau}{N} \ \ \ \ \ (10)$

Where ${\tau}$ is induce by

$\displaystyle \Gamma(N):=ker(SL_2({\mathbb Z})\rightarrow SL_2({\mathbb Z}/n{\mathbb Z})) \ \ \ \ \ (11)$

The key point is following:

Theorem 7 ${k={\mathbb C}}$ , the moduli of elliptic curves with full level n-structure is identified with

$\displaystyle \mu_N^*\times H/\Gamma(N) \ \ \ \ \ (12)$

Now we discuss the Mordell-Weil theorem.

Theorem 8 (Mordell-Weil theorem)

$\displaystyle E(F)\simeq {\mathbb Z}^r\oplus E(F)_{tor}$

The proof of the theorem divide into two part:

Weak Mordell-Weil theorem, i.e. ${\forall m\in {\mathbb N}}$ , ${E(F)/mE(F)}$ is finite.
There is a quadratic function,
$\displaystyle \|\cdot\|: E(F)\longrightarrow {\mathbb R} \ \ \ \ \ (13)$

${\forall c\in {\mathbb R}}$ , ${E(F)_c=\{P\in E(F), \|P\|<c\}}$ is finite.

Remark 5 The proof is following the ideal of infinity descent first found by Fermat. The height is called Faltings height, introduce by Falting. On the other hand, I point out, for elliptic curve ${E}$ , there is a naive height come from the coefficient of Weierstrass representation, i.e. ${\max\{|4a^3|,|27b^2|\}}$ .

While the torsion part have a very clear understanding, thanks to the work of Mazur. The rank part of ${E({\mathbb Q})}$ is still very unclear, we have the BSD conjecture, which is far from a fully understanding until now.

But to understanding the meaning of the conjecture, we need first constructing the zeta function of elliptic curve, ${L(s,E)}$ .

\paragraph{Local points} We consider a local field ${F_v}$ , and a locally value map ${F\rightarrow F_{\nu}}$ , then we have the short exact sequences,

$\displaystyle 0\longrightarrow E^0(F_{\nu})\longrightarrow E(F_{\nu})=E_s(\mathcal{O}_F)\longrightarrow E_s(K_0)\longrightarrow 0 \ \ \ \ \ (14)$

Topologically, we know ${E(F_{\nu})}$ are union of disc indexed by ${E_s(k_{\nu})}$ ,

$\displaystyle |E_s(k_{\nu})| \sim q_{\nu}+1=\# \mathop{\mathbb P}^1(k_{\nu})$

. Define ${a_{\nu}=\# \mathop{\mathbb P}^1(k_{\nu})-|E_s(k_{\nu})|}$ , then we have Hasse principle:

Theorem 9 (Hasse principle)

$\displaystyle |a_{\nu}|\leq 2\sqrt{q_{\nu}} \ \ \ \ \ (15)$

Remark 6 I need to point out, the Hasse principle, in my opinion, is just a uncertain principle type of result, there should be a partial differential equation underlying mystery.

So count the points in ${E(F)}$ reduce to count points in ${H^1(F_{\nu},E(m))}$ , reduce to count the Selmer group ${S(E)[m]}$ . We have a short exact sequences to explain the issue.

$\displaystyle 0\longrightarrow E(F)/mE(F) \longrightarrow Sha(E)[m] \longrightarrow E(F)/mE(F)\longrightarrow 0 \ \ \ \ \ (16)$

I mention the Goldfold-Szipiro conjecture here. ${\forall \epsilon>0}$ , there ${\exists C_{\epsilon}(E)}$ such that:

$\displaystyle \# (E)\leq c_{\epsilon}(E)N_{E/{\mathbb Q}}(N)^{\frac{1}{2}+\epsilon} \ \ \ \ \ (17)$

\paragraph{L-series} Now I focus on the construction of ${L(s,E)}$ , there are two different way to construct the L-series, one approach is the Euler product.

$\displaystyle L(s,E)=\prod_{\nu: bad}(1-a_{\nu}q_{\nu}^{-s})^{-1}\cdot \prod_{\nu:good}(1-a_{\nu}q_{\nu}^{-s}+q_{\nu}^{1-2s})^{-1} \ \ \ \ \ (18)$

Where ${a_{\nu}=0,1}$ or ${-1}$ when ${E_s}$ has bad reduction on ${\nu}$ .

The second approach is the Galois presentation, one of the advantage is avoid the integral model. Given ${l}$ is a fixed prime, we can consider the Tate module:

$\displaystyle T_l(E):=\varprojlim_{l^n} E[l^n] \ \ \ \ \ (19)$

Then by the transform of different embedding of ${F\hookrightarrow \bar F}$ , we know ${ T_{l}(E)/Gal(\bar F/F)}$ , decompose it into a lots of orbits, so we can define ${D_{\nu}}$ , the decomposition group of ${w}$ (extension of ${\nu}$ to ${\bar F}$ ). We define ${I_{\nu}}$ is the inertia group of ${D_{\nu}}$ .

Then ${D_{\nu}/I_{\nu}}$ is generated by some Frobenius elements

$\displaystyle Frob{\nu}x\equiv x^{q_{\nu}} (mod w),\forall x\in \mathcal{O}_{\bar Q} \ \ \ \ \ (20)$

So we can define

$\displaystyle L_{\nu}(s,E)=(1-q_{\nu}^{-s}Frob_{\nu}|T_{l}(E)^{I_{\nu}})^{-1} \ \ \ \ \ (21)$

And then ${L(s,E)=\prod_{\nu}L_{\nu}(s,E)}$ .

Faltings have proved ${L_{\nu}(s,E)}$ is the invariant depending the isogenous class in the follwing meaning:

Theorem 10 (Faltings) ${L_{\nu}(s,E)}$ is an isogenous ivariant, i.e. ${E_1}$ isogenous to ${E_2}$ iff ${\forall a.e. \nu}$ , ${L_{\nu}(s,E_1)=L_{\nu}(s,E_2)}$ .

$\displaystyle L(s,E)=L(s-\frac{1}{2},\pi ) \ \ \ \ \ (22)$

Where ${\pi}$ come from an automorphic representation for ${GL_2(A_F)}$ . Now we give the statement of BSD onjecture. ${R}$ is the regulator of ${E}$ , i.e. the volume of fine part of ${E(F)}$ with respect to the Neron-Tate height pairing. ${\Omega}$ be the volume of ${\prod_{v|\infty}F(F_v)}$ Then we have,

${ord_{s=1}L(s,E)=rank E(F)}$ .
${|Sha(E)|<\infty}$ .
${\lim_{s\rightarrow 0}L(s,E)(s-1)^{-rank(E)}=c\cdot \Omega(E)\cdot R(E)\cdot |Sha(E)|\cdot |E(F)_{tor}|^{-2}}$

Here ${c}$ is an explictly positive integer depending only on ${E_{\nu}}$ for ${\nu}$ dividing ${N}$ .

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Birch and Swinnerton-Dyer 猜想把椭圆曲线的有理点群和它的 $L$ 函数在 $s=1$ 处的零点阶联系起来。它是数论中最核心的桥之一：一边是 Diophantine 方程的解，另一边是解析函数的特殊值。

BSD 猜想入门：椭圆曲线、L 函数与 Mordell-Weil rank — BSD 猜想连接椭圆曲线有理点群的 rank 与 $L$ 函数在 $s=1$ 的零点阶。

1. Weierstrass 形式

在特征不是 $2,3$ 的域上，椭圆曲线常写成

$$E:y^2=x^3+ax+b,$$

并要求判别式

$$\Delta=-16(4a^3+27b^2)\ne0.$$

非零判别式保证曲线光滑。椭圆曲线不仅是代数曲线，它的点还带有 Abelian group 结构。

2. $j$-invariant 与 twist

$j$-invariant 分类椭圆曲线在代数闭包上的同构类：

$$j(E)=1728\frac{4a^3}{4a^3+27b^2}.$$

但在固定域 $K$ 上，同一个 $j$-invariant 可能对应不同的 twists。twist 说明“几何上同构”和“在基域上同构”之间有差别。

3. Mordell-Weil theorem

若 $E$ 定义在 $\mathbb Q$ 上，则 Mordell-Weil theorem 说

$$E(\mathbb Q)\cong E(\mathbb Q)_{\operatorname{tors}}\oplus\mathbb Z^r.$$

整数 $r$ 称为 Mordell-Weil rank。求 $r$ 是理解有理点结构的核心问题。

4. Hasse-Weil L 函数

对每个好素数 $p$，令

$$a_p=p+1-\#E(\mathbb F_p).$$

椭圆曲线的 $L$ 函数由 Euler product 组成：

$$L(E,s)=\prod_p(1-a_pp^{-s}+p^{1-2s})^{-1}$$

再在坏素数处加入修正因子。模性定理保证这个 $L$ 函数有解析延拓和函数方程。

5. BSD 猜想

BSD 猜想最核心的断言是

$$\operatorname{rank}E(\mathbb Q)=\operatorname{ord}_{s=1}L(E,s).$$

更精细的版本还给出 $L(E,s)$ 在 $s=1$ 处首项系数，涉及 regulator、Tate-Shafarevich group、Tamagawa numbers 和 torsion subgroup。

这个猜想的意义在于，它把有理点这个离散、代数的问题，转化成了 $L$ 函数特殊值这个解析问题。许多数论现代方法都在这座桥上来回移动。

分类： Elliptic curve

BSD 猜想入门：椭圆曲线、L 函数与 Mordell-Weil rank