Can manifolds be defined over fields other than $\mathbb{R}$ and $\mathbb{C}$?

Question

The question is one in the title. By other fields, I mean fields like $\mathbb{Q}$.

But I have never seen manifolds defined over fields other than $\mathbb{R}$ and $\mathbb{C}$. Even if one defines, would their study be as interesting as the theory of real or complex manifolds?

If there is a reference for this, please let me know!

Answer 1

A manifold is just something that locally looks like $\mathbb{R}^n$ ( + "making this make sense and behave reasonably" conditions).

But this can be all greatly extended and algebraic geometry gives the answer -- not only can you define them over other fields but you can have them be as crazy as you wish while still having some prescribed local properties.

Look up what a scheme, sheaf and (locally) ringed space is, for example.

Answer 2

See Banach manyfold and Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.