nt.number theory - How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?

Dear Maxmoo,

Just to offer a slightly different perspective than that given by Kevin and Brian:

While their advice is certainly correct, when I was learning this I also found it very
helpful to make a couple of "bare hands" computations, as a kind of reality check.

For this, begin with an elliptic curve in char. $2$, in fact with two, of the form:

$$y^2 + y = x^3$$

and

$$y^2 + x y = x^3 + x $$

One of these is supersingular, the other ordinary. (I won't tell you which here!)

Now try computing the $2$-torsion concretely, using lines passing through three points
and so on.

Remember that in the end you are looking for a degree $4$ equation (you may need to change
variables to see the point at infinity; this won't show up in the affine equations I've
given you). By general theory, you know this equation won't be separable: non-reduced
group scheme structure will show up concretely as inseparability in this polynomial.

In one case (the s.s. case) it will be entirely inseparable; in the other (ordinary) case
it will have inseparability degree $2$ (so "half" inseparable, "half" separable).

Once you've done the case of char. $2$, you might want to try char. $3$ as well (since
computing the equation for the 3-torsion is also just about in reach by hand).

The reason I suggest this is that I remember, when I was learning this stuff, that all
these group schemes (especially the non-reduced ones) seemed fairly ephemeral, but after
I had made these kind of explicit computations, I had a much more concrete mental model
for what the general theory was talking about, which gave me a lot more confidence in
reading and making arguments about these kinds of things.

Best wishes,

Matt