# SL_2(Z) and its congruence subgroups

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,

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

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

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

Remark 1

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

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

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

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,

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

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,

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

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

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

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 . \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)

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

${P=\phi(1,0), Q=\phi(o,1)}$ 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:

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

Given by,

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

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

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.

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

Where ${\tau}$ is induce by

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

The key point is following:

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

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

Now we discuss the Mordell-Weil theorem.

Theorem 8 (Mordell-Weil theorem)

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

The proof of the theorem divide into two part:

1. Weak Mordell-Weil theorem, i.e. ${\forall m\in {\mathbb N}}$, ${E(F)/mE(F)}$ is finite.
2. There is a quadratic function,

$\|\cdot\|: E(F)\longrightarrow {\mathbb R} \ \ \ \ \ (13)$

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

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

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

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

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

Theorem 9 (Hasse principle)

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

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.

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

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

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

\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.

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

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:

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

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

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

So we can define

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

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 ${\forall a.e. \nu}$, ${L_{\nu}(s,E_1)=L_{\nu}(s,E_2)}$.

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

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,

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

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