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

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

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

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

$$E/E \longrightarrow E$$

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

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

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

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."

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 $$E/E$$ well-defined, where I've assumed $$E \cong \mu_{3} \times \mu_{3}$$? 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 $$E/E$$ by studying $$\mu_{3} \times \mu_{3}$$-torsors over $$\text{Spec}(\mathbb{F}_{p^{n}})$$?

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

1. $$E$$ 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, $$E(\overline{\mathbb{F}_3})$$ should contain a non-trivial point (but $$(\mu_3 \times \mu_3)(\overline{\mathbb{F}_3})$$ does not).

Even if $$E$$ is supersingular, $$E$$ 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 $$K=E$$ 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.)

• Why should an ordinary elliptic curve have a 3-torsion point over $\mathbf F_3$? You mean $\overline{ \mathbf F_3}$? Also I feel like the spirit of the question is more about $p\ne 3$, nice answer though! Nov 17 '18 at 22:47
• Thanks for the answer! I believe I understand that $\mu_{3}$ might not consist of three distinct points. It's really the scheme $\text{Spec}k[x]/(x^3-1)$, so depending on $k$, it might have fewer than three points, but with fattening. Is this not consistent with saying that not every elliptic curve has 9 distinct 3-torsion points? Perhaps I'm misunderstanding something. Nov 17 '18 at 23:08
• @Alex Oh yes, you're absolutely right about $\overline{\mathbb{F}_3}$! I guess when $p\neq 3$ and $k$ is algebraically closed, the intuition from $\mathbb{C}$ transfers nicely.
– dyf
Nov 17 '18 at 23:10
• @Benighted Ooops you are right, i guess I overlooked the word "not"... sorry for the mistake!
– dyf
Nov 17 '18 at 23:11