TITLE: Geometry Vs Arithmetic of schemes
QUESTION [11 upvotes]: Let's suppose we have a Scheme $X$ over the the field $k$, where such a field can be though to be either $\mathbb{C}$ or a finite field $\mathbb{F}_q$. Then having this in mind, 
Where do we find some representative examples where Geometry governs arithmetic? That is to say, examples where the geometry (or topology) of $X$ over $\mathbb{C}$ dictates the arithmetic behavior over $\mathbb{F}_q$.
Answers along with references would be highly appreciated.

REPLY [2 votes]: Look at Darmon's article on "Arithmetic of Curves".