I've heard people say that (part of) algebraic geometry deals with generalizing results from differential geometry to fields other than $\mathbb{C}$. Some searching has brought me to "GAGA Theorems", but I don't have anywhere near the background that it looks like I might need to understand those. It seems like this program was started in order to work on the Weil Conjectures, since the computational evidence seemed to point to "continuous" theorems (such as the Lefschetz fixed point theorem) being relevant in a "discrete" setting. Again, though, I don't have the background to understand even the wikipedia page.
I have taken a class in "classical" algebraic geometry (using Shaferevich, Vol 1), but it is still unclear to me which aspects of differential geometry were able to be transported, and how. The only analogies that I can see for myself are
the definition of an abstract variety (which we barely touched on at the end of my class) being clearly inspired by the definition of an abstract manifold
the notion of "singular" and "regular" points on a curve, which are also clearly inspired by their analogues in differential geometry.
I would appreciate it if someone can provide some insight into this program at a level I can understand. I am perfectly happy to see some hand-waving if that helps, but I would also like to see (relatively) concrete examples of ideas in differential geometry being moved to the world of arbitrary fields.
Thanks!