Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ be a prime where $E$ has good reduction. Is there always a curve $C$ defined over $\mathbb{Q}$ such that $\# J(\mathbb{F}_p) = \#E(\mathbb{F}_p)^2$ or $\# J(\mathbb{F}_p) = \#E(\mathbb{F}_p)^3$ ($J$ is the Jacobian of $C$)?
After some computations involving the trace of Frobenius on the etale cohomology groups of $C$ and $C \times C$, it seemed like $\#C(\mathbb{F}_p)$ must reach the upper or lower bound given by the Hasse-Weil bound in order to have an isomorphism $J(C)\cong E^g$. However, this is impossible since $p$ is not a square. This sounded strange to me for the following reason:
By the Torelli theorem, $E^2$ and $E^3$ must be isomorphic to the Jacobian of some curve over an algebraically or separably closed field. I wasn't sure about what the situation was over an arbitrary field, but there seems to be a more general version according to the discussion in the following question: When does the Torelli Theorem hold?
Is there something wrong with the computation made above or is there something I'm missing about properties of the ground field which implies that the situation mentioned above is possible? In addition, what happens in the case of number fields (taking the residue field at a prime)? Is there anything new that happens in that case?
