nt.number theory - Global fields: What exactly is the analogy between number fields and function fields?

Sure, here's a overview.

Suppose you have a ring R over a field k, then, by the magic of algebraic geometry, you can think about it in a geometric way. You do this by defining points as epimorphisms R to k and finding out that a lot of geometric intuition plays out nicely.

Now if you start with a field, the above procedure gives you just a single point, so it's more interesting to find ring inside it — people usually take the ring of integers inside the field, which is uniquely defined.

Now the amazing thing is that you can perform this exact procedure either on number fields like Q or on function fields like F_p(t) and it gives you a very similar geometric structure.

For example, you can talk about completion of your ring by some maximal ideal and this corresponds to considering infinitesimal geometry around a single point. For number fields that would be something like Q_p while for function fields that would be F_p[[t]]. Not if you think how Q_p is basically F_p formally extended by p you notice the techniques wors the same in both cases.

E.g. the theory of ramification is basically the theory of extending either F_p[[t]] or Q_p. (There are important differences though — F_p[[t]] can be extended with F_{p^2}[[t]])