Let $C$ be a smooth genus $g>1$ curve defined over a number field (say over $\mathbb Q$). Let $J(C)$ be its jacobian. Suppose that $J(C)$ has good reduction $J_p$ at a prime $p$ and moreover $J_p$ is absolutely simple (ie simple over any field extensions).

Is it true that $C$ has good reduction $C_p$ at $p$ (maybe after some finite extension) ? So far in the (few) examples I found in the literature of a curve that has bad reduction and for which the jacobian has good reduction, that reduction is isogeneous to a (non-trivial) product.

A question related to that problem is : what is the link between $C_p$ and $J_p$ ? Is there still a non-constant map $C_p \to J_p$ ? Does the image of that map generates $J_p$ ?

There is a criteria of Oda for good reduction of curves mentioned here: A curve with bad reduction for which the jacobian has good reduction but I do not see how to apply it to the hypothesis I make on the curve $C$...