I am reading this paper, specifically Example 2.3 on page 9, and am having a few problems understanding a part of it
We construct an elliptic curve $E$ on $\mathbb{F}_{11}$ defined by $y^2=x^3+4x$ with a point at infinity $\mathcal{O}$
We then find a divisor of $E$ $$D=\left[(0,0)\right]+\left[(2,4)\right]+\left[(4,5)\right]+\left[(6,3)\right]-4\left[\mathcal{O}\right]$$
We can then say that: $$\begin{align}\text{div}(y-2x)&=\left[(0,0)\right]+2\left[(2,4)\right]-3\left[\mathcal{O}\right]\\ \text{div}(x-2)&=\left[(2,4)\right]+\left[(2,-4)\right]-2\left[\mathcal{O}\right]\end{align}$$
We can then replace various elements in the definition of $D$ with these divisors:
$$\begin{align}D&=\left[(0,0)\right]+\left[(2,4)\right]+\left[(4,5)\right]+\left[(6,3)\right]-4\left[\mathcal{O}\right]\\ &=\left[(2,4)\right]+\text{div}\left(\frac{y-2x}{x-2}\right)+\left[(4,5)\right]+\left[(6,3)\right]-3\left[\mathcal{O}\right]\\ &=\left[(2,-4)\right]+\text{div}\left(\frac{y-2x}{x-2}\right)+\left[(2,4)\right]+\text{div}\left(\frac{y+x+2}{x-2}\right)-2\left[\mathcal{O}\right]\\ &=\text{div}\left(x-2\right)+\text{div}\left(\frac{y-2x}{x-2}\right)+\text{div}\left(\frac{y+x+2}{x-2}\right)\end{align}$$
However, what I don't understand are the following steps:
$$\begin{align} D&=\text{div}\left(x-2\right)+\text{div}\left(\frac{y-2x}{x-2}\right)+\text{div}\left(\frac{y+x+2}{x-2}\right)\\ &=\text{div}\left(\frac{(y-2x)(y+x+2)}{x-2}\right)\\ &=\text{div}\left(x^{2}-y\right)\end{align}$$
Can anyone help explain to me how they have come to this conclusion please
This question comes in 3 parts:
- Step 1 - Elliptic Curve and Divisor Example help (Step 1)
- Step 2 - Elliptic Curve and Divisor Example help (Step 2)
- Step 3 - This one
Conclusion:
$$\begin{align} D&=\text{div}\left(x-2\right)+\text{div}\left(\frac{y-2x}{x-2}\right)+\text{div}\left(\frac{y+x+2}{x-2}\right)\\ &=\text{div}\left((x-2)\left(\frac{y-2x}{x-2}\right)\left(\frac{y+x+2}{x-2}\right)\right)\tag{div$(f\cdot g)=$ div$f+$ div$g$}\\ &= \text{div}\left(\frac{(x-2)(y-2x)(y+x+2)}{(x-2)^2}\right)\tag{multiply out}\\ &= \text{div}\left(\frac{(y-2x)(y+x+2)}{x-2}\right)\tag{cancel $(x-2)$}\\ &= \text{div}\left(\frac{-2x^2-xy-4x+y^2+2y}{x-2}\right)\tag{multiply out}\\ &= \text{div}\left(\frac{-2x^2-xy-4x+\left(x^3+4x\right)+2y}{x-2}\right)\tag{substitute $y^2=x^3+4x$}\\ &= \text{div}\left(\frac{-2x^2-xy+x^3+2y}{x-2}\right)\tag{simplify numerator}\\ &=\text{div}\left(x^2-y\right)\tag{simplify fraction} \end{align}$$