The 3-torsion Points of an Elliptic Curve over Finite Fields

Question

Let $E/\mathbb{C}$ be a smooth elliptic curve, and let $E[3]$ be the 3-torsion points. We have a non-canonical isomorphism

$$ E[3] \cong \mathbb{Z}/3 \times \mathbb{Z}/3$$

and if I'm not mistaken, the multiplication by 3 map gives an isomorphism

$$E/E[3] \longrightarrow E$$

of $E$ with its quotient by the 3-torsion points.

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In general, most of my intuition comes out of complex algebraic geometry, and I tend to struggle even with basic things over finite fields. So my first question is the following:

If we now have $E/\mathbb{Q}$ a smooth elliptic curve, and we reduce modulo $\mathbb{F}_{p^{n}}$ such that $E/\mathbb{F}_{p^{n}}$ is smooth, should we really think of the 3-torsion points as the group scheme

$$E[3] \cong \mu_{3} \times \mu_{3}?$$

In other words, not all elliptic curves have 9 distinct 3-torsion points, correct? I presume they will over the algebraic closure, but otherwise there might be "fattening."

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Basically, I was hoping to use this simple setting as a testing ground for how to count points on quotient varieties over $\mathbb{F}_{p^{n}}$ (which seems to me to be quite a delicate issue).

So over $\mathbb{F}_{p^{n}}$, to what extent is $E/E[3]$ well-defined, where I've assumed $E[3] \cong \mu_{3} \times \mu_{3}$? Do we still have an isomorphism with $E$ itself? Perhaps this should be a question of its own, but can one think of counting points on $E/E[3]$ by studying $\mu_{3} \times \mu_{3}$-torsors over $\text{Spec}(\mathbb{F}_{p^{n}})$?

Answer 1

When it comes to schemes, I think it's the best to specify it's a scheme over what. For example, $E[3]$ over $\mathbb{F}_3$ is never $\mu_3 \times \mu_3$, and there are a few ways to see this:

1. $E[3]$ is self-dual (in the sense of Cartier duality in the theory of finite group schemes), and over characteristic $3$, the dual of $\mu_3$ is $\mathbb{Z}/3\mathbb{Z}$, and these two group schemes are not isomorphic, since the former has $1$ $\mathbb{F}_3$-point, and the latter has $3$. (You also said "not all elliptic curves have $9$ distinct $3$-torsion points", which suggests that you think $\mu_3$ always have $3$ points, but this is not right!)
2. $E$ can be either ordinary or supersingular, and if it's ordinary, $E[3](\overline{\mathbb{F}_3})$ should contain a non-trivial point (but $(\mu_3 \times \mu_3)(\overline{\mathbb{F}_3})$ does not).

Even if $E$ is supersingular, $E[3]$ as a group scheme is still never $\mu_3 \times \mu_3$, because of the first point! (In fact it's not easy to see what this really is, but this is a story for another day.)

Towards the second part, you mentioned quotient varieties. I am no algebraic geometer, so to me the word "quotient" already rings a huge alarm in me, and I shall humbly leave this part for others to answer. (I might not be right on this, but I would like to think that since $K=E[3]$ is finite, $E/K$ is still nicely and well defined and the isomorphism is still fine.)

(Editted according to Alex's and OP's comment.)