Manifold over a Finite Field

Question

I'm trying to either associate a manifold with a finite field, or, ideally find a way of considering finite fields as manifolds, in a non-trivial manner.

I hope to be able to use this to extend topological methods to finite projective planes.

Answer 1

Look for the spectrum of the finite field you are considering. It is an affine scheme whose underlying topological space is a singleton, but which is carrying information through its sheave of rings of functions.

Answer 2

Probably what you're looking for is the étale cohomology (and other étale versions of classical topological material) of the projective spaces over finite fields, which can be regarded as schemes. Schemes can be regarded as a natural (though quite vast) generalization of complex projective manifolds.