Vrftjkry

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.

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."

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 $text{Spec}(mathbb{F}_{p^{n}})$?

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 $mathbb{Z}/3mathbb{Z}$, 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.)