twists of elliptic curves over finite fields

Question

Let $p\ge 5$ be prime. For every $j\in\mathbb{F}_p$, there are at most 6 twists of any elliptic curve over $\mathbb{F}_p$ with $j$-invariant $j$, and in general only two twists.

Is there a formula which gives the point counts $\#E(\mathbb{F}_p)$ (equivalently, the traces of Frobenius $a_p(E)$) of the finitely many elliptic curves over $\mathbb{F}_p$?

In particular, if $j\in\mathbb{F}_p$ and $j\not\equiv 0,1728\mod p$, then there are precisely two nonisomorphic elliptic curves $E_j,E_j'$ over $\mathbb{F}_p$ with $j$-invariant $j$. Is there a relation between $a_p(E_j),a_p(E_j')$?

Answer 1

If $p>2$ and $j \neq 0,1728$ then the two twists are of the form $E: y^2=f(x)$ and $E_d: d{y}^2=f(x)$ with $d \notin {\mathbb{F}_p^{*}}^{2}$, so $a_{E_d} = -a_E$, as follows: Denote $\chi(0) = 0$, $\chi(x) = 1$ if $x \in {\mathbb{F}_p^{*}}^{2}$, $\chi(x) = -1$ otherwise. Then $\#{E(\mathbb{F}_p)} = 1 + \Sigma_{x \in \mathbb{F}_p} (1+\chi(f(x)))$ so $a_E= - \Sigma \chi(f(x))$, $a_{E_d}= - \Sigma \chi(f(x)/d) = -a_E$.

Answer 2

As David said, the two quadratic twists over $\mathbb F_p$ satisfy $a_p(E_1)+a_p(E_2)=0$. This is also easy to see if you think of taking the trace of the action of Frobenius on $T_\ell(E)$ and on the twist.

You also ask about the values of $a_p(E)$ as $E$ varies over all of the $\mathbb F_p$. That's more interesting. The answer is that as $p\to\infty$, the distribution is the one that appears in the Sato-Tate conjecture (now a theorem for elliptic curves over $\mathbb Q$). Thus writing $a_p(E)=2\sqrt{p}\cos\theta_p(E)$, the set of values $\{\theta_p(E) : E/\mathbb F_p\}$ approaches a $\sin^2$ distribution as $p\to\infty$. This is a theorem of Birch:

B. J. Birch, MR 230682 How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc. 43 (1968), 57--60.