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 supersingular curves.)

(My example curve $E$ is never smooth in characteristic $2$, but if you consider a more general Weierstrass form which is smooth in char. $2$, then you can find a degree $1$ curve $C$ which does what I ask. For instance, if $E: y^2z+A\,xyz+yz^2=x^3$, take $f=A\,x+2\,y+z$.)

So my questions are:

1. Is it usually possible to find an equation $f=0$ such that $E\cap C$ is exactly the $p$-torsion? (Is this the same as asking that $X$ is a complete intersection?) Can you ever show it's not possible?
2. Are there known methods for computing the locus of $p$-torsion points explicitly? Are there software packages which do this? (I'm aware there are ways to find explicit torsion points on elliptic curves defined over some field or number ring; I'm asking for something a little different, I think.)
3. Have people carried out these sorts of computations for various small values of $p$ (even $p=5$), and are these computations described in print? (I'm probably most interested in this question.)

Warning: I am not an algebraic geometer or number theorist.

1

# How do you explicitly compute the p-torsion points on a general elliptic curve in Weierstrass form?

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 $\mathbb{Z}/p\to E$".

(My example curve $E$ is never smooth in characteristic $2$, but if you consider a more general Weierstrass form which is smooth in char. $2$, then you can find a degree $1$ curve $C$ which does what I ask. For instance, if $E: y^2z+A\,xyz+yz^2=x^3$, take $f=A\,x+2\,y+z$.)

So my questions are:

1. Is it usually possible to find an equation $f=0$ such that $E\cap C$ is exactly the $p$-torsion? (Is this the same as asking that $X$ is a complete intersection?) Can you ever show it's not possible?
2. Are there known methods for computing the locus of $p$-torsion points explicitly? Are there software packages which do this? (I'm aware there are ways to find explicit torsion points on elliptic curves defined over some field or number ring; I'm asking for something a little different, I think.)
3. Have people carried out these sorts of computations for various small values of $p$ (even $p=5$), and are these computations described in print? (I'm probably most interested in this question.)

Warning: I am not an algebraic geometer or number theorist.