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 a references would be highly appreciated.

Let

Let's suppose we have an a Scheme $X$ over the the field $k$ 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 a references would be highly appreciated.

Let suppose we have an Scheme $X$ over the the field $k$ where such a field can be though to be $\mathbb{C}$ 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, where the geometry (or topology) of $X$ over $\mathbb{C}$ dictates the arithmetic behavior over $\mathbb{F}_q$.

Answers along with a references would be highly appreciated.