What is this notation? $V(\Bbb Z/p\Bbb Z)$ I'm trying to write a blog post, and I've run into a stumbling block with notation.

Is $V(\Bbb Z/p\Bbb Z)$ a standard notation in algebraic number theory? Does it mean a variety restricted to the integers mod $p$? If not, what does it mean / could it mean in this context?

Context: I attended a lecture recently on Strong Approximation in number theory. The speaker began by talking about Markoff's surface
$$x^2+y^2+z^2-3xyz=0,$$
and told us about a group $\Gamma$ which generates the integral solutions of this equation. He then made a conjecture, which didn't seem to be tied to Markoff's surface in particular:


*

*Conjecture: The action of $\Gamma$ on $X(\Bbb Z_p)$ has precisely two orbits, one of which is $\{0\}$.


After digging around online, I found notes for another talk he had given. On page 12, he defines $X^*(p)=V(\Bbb Z/p\Bbb Z)\smallsetminus\{0\}$, so I'm assuming my $X$ is that $X^*$ but with $\{0\}$ included. 
He defines "$V$" as the Zariski closure of an orbit $O\subseteq \Bbb Z^n$ on page 3, but I'm not sure it's the same $V$.

This is common notation in algebraic geometry. I have two schemes $X,T$ with structure morphisms to another scheme $S$, and now $X(T)$ is the set of $S$-morphisms $T \to X$. When $T$ is the spectrum of some ring $R$ we usually just write $X(R)$.
In this situation, $S = \mathbb Z$ (we really could have left this bit out here, since ring homomorphisms always respect $\mathbb Z$), 
\[
X = \operatorname{Spec} \mathbb{Z}[x,y,z]/(x^2+y^2+z^2-3xyz),
\]
 and $T = \operatorname{Spec} \mathbb Z_p$. Since everything is affine, morphisms $T \to X$ correspond to ring homomorphisms
\[
\mathbb{Z}[x,y,z]/(x^2+y^2+z^2-3xyz) \to \mathbb Z_p;
\]
in other words, solutions $(x,y,z)$ to your equation in $\mathbb Z_p$. We've said this simple thing in a complicated way using Grothendieck's functorial language, but I do want to point out (a) how natural this is (b) how nice it is to have a geometric object like $X$ that already "knows" how its defining equations might be solved, in the sense that the scheme $X$ and the collection of all the $X(T)$ determine each other.
