Take the 2-minute tour ×  Consider the Weierstrass cubic $$y^2z
= x^3 + A\, xz^2+B\,z^3.$$ This defines a curve $E$ in
$\mathbb{P}^2$, which if smooth is an elliptic curve with
basepoint at $[0,1,0]$.  I'm interested in having an explicit
description of the locus of $p$-torsion points of this curve,
where $p$ is prime.  In fact, suppose $p\neq 3$. Then ideally
I'd like to be able to find a curve $C$ in $\mathbb{P}^2$, given
by an equation $f=0$ of degree $d=(p^2-1)/3$, so that the scheme
$X=E\times_{\mathbb{P}^2} C$ is precisely the locus of points of
exact order $p$.  Example: For $p=2$, it's well known that $f=y$
gives such a curve.  I'd like $f$ to be an expression which
depends on $A$ and $B$; i.e., I want to do this over a generic
part of the moduli stack. I would also like this expression to
work in characteristic p; in this case, $X$ should turn out to
be the "scheme representing Drinfeld level structures
$\mathbb{Z}/p\to E$". (Edit: I'm particularly interested in
families of curves which include super