Part I of Explicit Construction of the Elliptic Curve (CIMPA School)

08:53:00   Algebraic

The following is based on Stevenhaagen's talk on 2013 CIMPA School in Manila. This write is a little bit mess, as actually a lot of things need to be proved. I'm just trying to connect the preliminaries introduced so far.

Recall from my previous post  a Weierstrass Elliptic Function is given by

$$\wp(z)=\frac{1}{z^2} + \sum_{\omega \in A/\{0\}} \left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right)$$

By the doubly periodicity, let's take the restriction of the domain inside a fundamental lattice $L$ .  This function also satisfies the differential equation

$${\wp^{\prime}(z)}^2 = 4 \wp(z)^3-g_2(L)\wp(z)-g_3(L)$$

where $g_1(L)$ and $g_2(L)$ are modular forms given by $g_2(L)=60 \sum_{\omega \in L/\{0\}} \frac{1}{\omega^4}$ and $g_3(L)=140 \sum_{\omega \in L/\{0\}} \frac{1}{\omega^6}$.

Therefore we can define the parametrization of a torus $T:=\mathbb{C} \pmod L$ to a Projective Plane by the equation of  Elliptic curve
$$y^2=4x^3-g_2(L)x-g_3(L)$$
with dehomogenization $$y^2z=4x^3-g_2(L)xz^2-g_3(L)z^3$$

The map is given by $z \mapsto (\wp(z),\wp^{\prime}(z),1)$  and $0 \mapsto (0,1,0)$, this map is an injective function from $T$ to Elliptic Curve.

We also define the $j$-invariant of the Latice L by 

$$j(L)=1728\frac{g_2(L)^3}{g_2(L)^3-27g_3(L)^2}$$

this value is always defined whenever $L$ is indeed  a lattice ($g_2(L)^3-27g_3(L)^2 \neq 0$).

Now for a complex number $\tau$ in a Upper Half plane $\mathcal{H}$, define the lattice $L_{\tau}$ as the lattice generated by $1$ and $\tau$, we write $L_{\tau}=[1,\tau]$.  It can be shown that every lattice $L=[z_1,z_2]$ is isomorphic (or sometimes' called homothetic) to $L_{\tau}$  for some $\tau \in \mathcal{H}$ (this can be seen by setting $\tau=\frac{z_2}{z_1}$). Therefore we can view $j$ as the function $j : \mathcal{H} \rightarrow \mathbb{C}$ with

$$j(\tau)=j(L_{\tau}) = 1728\frac{g_2(L_{\tau})^3}{g_2(L_{\tau})^3-27g_3(L_{\tau})^2}$$

We call this function the $j$-function.  The key point is that this function is a modular form of weight 0, i.e modular function, i.e

$$j(A \cdot \tau) = j(\tau)$$

for $A \in SL_2(\mathbb{Z})$, and $A \cdot \tau$ denotes the Mobius action.

The condition that $\tau$ is on the Upper Half Plane is good for redefine the function $j$ on the smaller subset of $\mathcal{H}$ which is a  fundamental domain $\mathcal{D}$ given by

$$\mathcal{D}=\left\{\tau \in \mathcal{H} \, \lvert \, \Re(\tau) \in \left[-\frac{1}{2},\frac{1}{2}\right), \tau > 1 \right\} \cup \{\tau \in \mathcal{H} \, \lvert \, \Re(\tau) \geq 0 \mbox{ if } |\tau|=1 \}$$  

this fundamental domain will correspond each $\tau \in \mathcal{D}$ into $\tau=\frac{-b+\sqrt{b^2-4ac}}{2a}$, where $a,b$ and $c$ are coprime integer satisfying 

$$|b|\leq a \leq c \text{ and } b\geq 0  \text{ whenever } |b|=a \text{ or } a=c$$

We call the function $f(x,y)=ax^2+bxy+cy^2$ a reduced quadratic form whenever $(a,b,c)$ satisfying the above conditions. Therefore $\tau \in \mathcal{D}$ whenever $\tau$ is the (half-plane) root of $f(x,1)$ for a reduced quadratic form $f(x,y)$.

Now given a negative integer $D$,  we collect all the reduced quadratic form $f(x,y)=ax^2+bxy+cy^2$ such that $D=b^2-4ac$,  it turns out that this collection say $Cl(D)$ is a class group. More interesting is that we can adjoint $j(\tau)=j\left(\frac{-b+\sqrt{b^2-4ac}}{2a}\right)$ to $K=\mathbb{Q}(\sqrt{D})$ where $D$ is a fundamental discriminant, such that $j(\tau)$ is algebraic in this extension!! We called this extension a Hilbert Class Field, and the minimum polynomial $H_K$ is called Hilbert Class Polynomial and it is in $\mathbb{Z}[X]$. 

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