Let $p$ be a prime number. A global field is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. On the other hand one can show that a local field (which is by definition a complete discrete valuation field with finite residue field) is a finite extension of $\mathbb Q_p$ or $\mathbb F_p((t))$.
Why do we use the adjectives "local" and "global"? What is the geometric picture that I am missing?
Best Answer
In the early 1930s when local class field theory was developed, it was always called "class field theory in the small" (in German). Eventually the term local prevailed. In 1935, Hasse wrote, in a review of an article of Schilling, about the "local arithmetic structure of algebras". Later on, especially when articles in French and English on class field theory began appearing, the expression "local" was used, and the dichotomy small-large became local-global. In particular Chevalley's articles on local class field theory from 1938 use "theorie locale des corps de classes".
I don't think there's a big picture you're missing: in local fields, you always study one prime ideal; in global fields, you look at all of them at the same time. As a matter of fact, global fields can be characterized by a product formula (Artin-Whaples etc.).