I am reading some papers about elliptic curves and I come across the term $l$-th division polynomial of $E$. I don't really know much about field theory and I tried to look for this definition but I'm unable to find it. I'm wondering if anyone can provide a high level explanation?
Sorry if this question is too basic.
If one has an elliptic curve $E$ in Weierstrass form $$y^2=x^3+ax^2+bx+c$$ and take a typical point $P=(x_1,y_1)$ and consider the multiples of it in the elliptic curve group $[n]P=P+\cdots+P=(x_n,y_n)$ then $$x_n=\frac{\phi_n(x_1)}{\psi_n(x_1)^2}$$ where $\phi_n$ and $\psi_n$ are polynomials. Then the $\psi_n(x)$ are the division polynomials: $[n]P=O$ iff $\psi_n(x_1)=0$.
This isn't strictly true. When $n$ is odd this works fine. When $n$ is even $\psi_n(x_1)$ isn't really a polynomial in $x_1$, it actually equals $y_1$ times a polynomial in $x_1$, but as $y_1^2=x_1^3+\cdots$ then $\psi_n(x_1)^2$ is a polynomial in $x_1$.
There's a good account of all this in Washington's book on elliptic curves.