Let E be an elliptic curve defined over $\mathbb{Q}$ (coeffs. there), and consider its $n-$torsion points in $\mathbb{C}$, $E(\mathbb{C})_{\text{tors}}[n]$. We know this group is isomorphic to $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$. Moreover, if we consider $K_n$ to be the field extension of $\mathbb{Q}$ obtained by adjoining the coordinates of the $n$-torsion points we obtain a natural action of $\text{Gal}(K_n / \mathbb{Q})$ on $E(\mathbb{C})_{\text{tors}}[n]$, so we obtain an injective representation
\begin{equation*}
\chi_n : \text{Gal}(K_n / \mathbb{Q}) \hookrightarrow \text{Aut}(E(\mathbb{C})_{\text{tors}}[n]) \cong \text{GL}_2(\mathbb{Z}/n\mathbb{Z})
\end{equation*}
If we consider a prime $p$ of good reduction for $E$, and let $\text{Frob}_p$ be the corresponding Frobeniu element in $\text{Gal}(K_n/\mathbb{Q})$ (I know the Frobenius elements constitute, in fact, a conjugacy class $\mathcal{C}_p$: pick any of them for our purposes), then there is a theorem (in Silverman, for example) which says that
\begin{equation*}
\begin{cases}
\text{det}( \chi_n(\text{Frob}_p)) \equiv p \text{ (mod }n) \\
\text{Tr}(\chi_n(\text{Frob}_p)) \equiv a_p \text{ (mod }n)
\end{cases}
\end{equation*}
where $a_p = 1+p-|E(\mathbb{F}_p)|$.
I want to apply this theorem for the elliptic curve $y^2 = x^3 – 1$; in particular I want to say something about $a_p$ for a given $p$. I think there is some result which states
1) If $p \equiv 2 \text{ (mod }3)$, then $a_p = 0$.
2) If $p \equiv 1 \text{ (mod }3)$, then $a_p = 2a$ where $a$ is such that $p = a^2+ab+b^2$ (or something similar).
Can someone give some reference on where to find such a result, or where I can find a similar study for elliptic curves of the form $y^2 = x^3 + D$? My effords from now on have been simply computing the trace by hand for some specific cases ($p=2$ and $3$), but it gets extremely tedious even for $p=3$.
Thank you very much!
Best Answer
The key fact you need is this: if $E/K$ is an elliptic curve which has complex multiplication over $K$, then the associated $\ell$-adic Galois representations $$\rho_{E,\ell}:G_K\to\mathrm{GL}_2(\overline{\mathbb Q}_\ell)$$ are reducible for all primes $\ell$. Indeed, $\mathrm{End}(E)\otimes\overline{\mathbb Q}_\ell$ embeds into the endomorphism ring of $\rho_{E,\ell}$, and the former is not a division ring.
In your case, your curve $E: y^2 = x^3-1$ does not have CM over $\mathbb Q$, but it picks up extra endomorphisms over $K=\mathbb Q(\zeta_3)$. It follows that $\rho_{E,\ell}$ is irreducible, but $\rho_{E,\ell}|_{G_K}$ becomes reducible, and representation theory kicks in to show that $$\rho_{E,\ell}\cong \mathrm{Ind}_{G_K}^{G_\mathbb Q}\chi_\ell\qquad(*)$$ where $\chi_\ell$ is a character -- actually it is the Galois character corresponding to the Hecke grossencharacter which is attached to $E$ by the theory of CM.
Finally, one can show using pure representation theory that $(*)$ is equivalent to the statement $$\rho_{E,\ell}\cong\rho_{E,\ell}\otimes\theta\qquad(**)$$ where $\theta:G_\mathbb Q\to\overline{\mathbb Q}_\ell^\times$ is the one dimensional representation which has as its kernel $G_K$. Explicitly, $\theta$ is a lift of the unique non-trivial Dirichlet character of conductor $3$.
In particular: