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:
- 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?
- 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.)
- 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.
Best Answer
http://en.wikipedia.org/wiki/Division_polynomials
That's not the best wikipedia page. "The division polynomials form an elliptic divisibility sequence." is mentioned well before the far more important "the roots of the n'th division polynomial tell you the n-torsion in the curve".