You might be interested in Serre's recent book
Lectures on $N_X(p)$, AK Peters, Taylor and Francis, New York, 2011, 163 pages,
a version of which is available on his website at the Collège de France.
Serre deals with the following basic question : Let $X$ be a scheme of finite type over $\mathbf{Z}$. What can you say about the number $N_X(p)$ of points of $X$ over the finite field $\mathbf{F}_p$, as $p$ varies (over the primes) ? For example, it is not entirely trivial (Theorem 1.1) to show that $X$ is empty if and only if $N_X(p)=0$ for all sufficiently large $p$.
Another aspect of the question is the relationship between the analytic space $X(\mathbf{C})$ and the sequence $N_X(p)$. For example, can you recover the dimension of $X(\mathbf{C})$, or the number of irreducible components of $X(\mathbf{C})$ from the sequence $N_X(p)$ ? See Theorem 1.2 for this.
Later (Thoerem 6.15) he discusses what conclusion can be drawn if $|N_X(p)-N_Y(p)|<2$ for a set of $p$ of density $1$.
