A Jacobian with a good reduction, which is simple : how is the reduction of the curve? 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 $C$...
 A: Yes, this is true. One can argue as follows:
Since $J(C)$ has good reduction, your other hypothsism implies that the special fibre of its Neron model at $p$ is an absolutely simple abelian variety. By Grothendieck's theorem, $C$ has semi-stable reduction at $p$, so it has a regular model whose fibre at $p$ is a semi-stable curve. 
By the last line of Theorem 4, p. 267, of the book "Neron Models" by Bosch, Raynaud and Luetkebohmert, the identity component of the Neron model of $J(C)$ is $\mathrm{Pic}^0$ of the  regular semi-stable model of $C$. (By a base change one can assume that all components of the special fibre are geometrically irreducible.) 
By Example 8 on p. 246 of the same book, which gives the structure of $\mathrm{Pic}^0$ of a semi-stable curve over a field, and your assumptions on the reduction of $J(C)$, it follows that $C_p$, the special fibre of a regular semi-stable model of $C$, must have a unique smooth component of genus $g(C)$ and possibly some other smooth rational components.
Since the arithemetic genus is constant in flat families it follows that any extra rational component must be a "tail" and  can be contracted. We conclude that the special fibre of the stable model of $C$ is a smooth curve of the same genus, i.e., $C$ has good reduction at $p$.
(Note that no base change is required to get good reduction.) 
