Let $M_{g} \to \mathbb{Z}$ be the coarse moduli scheme of non-singular genus g curves over $\mathbb{Z}$. That is, suppose that $M_{g}$ co-represents the functor $M^{\sharp}_{g} : \text{Sch} \to \text{Sets}$ whose Yoneda points are (flat) families of non-singular, genus $g$ curves.  


**What is $M_{g} \times \mathbb{Z}/p$?**


One would expect that $M_{g} \times \mathbb{Z}/p$ co-represents $M^{\sharp}_{g} \times \mathbb{Z}/p$, but in general, the formation of GIT quotients can [fail](http://mathoverflow.net/questions/38529/uniform-quotient-vs-universal-quotient) to commute with passing to fibers. Does that failure occur here?

In other words: **does $M_{g} \times \mathbb{Z}/p$ co-represent $M^{\sharp}_{g} \times \mathbb{Z}/p$?**




Some references for the construction of $M_g$ over $\mathbb{Z}$ can be found [here](http://mathoverflow.net/questions/65932/moduli-space-of-curves-over-mathbb-z).

*Added:*  Torsten Ekedahl had concerns about the use of the term ``co-represents."  I intended it to mean that there is a natural transformation $M^{\sharp}_{g} \to M_g$ that is universal with respect to transformations into a scheme.   Of course, this should be the same as asking that $M_{g}$ is the coarse space of the moduli stack $\mathcal{M}_{g}$, for a suitable notion of coarse space.