Results like this are called "local-to-global" results, since they're analogous to constructing a function from local information, like its Laurent series at all different points.
In 1950, in his famous PhD thesis, John Tate came up with a clever way to formalize this "Laurent series at all different points" idea in the context of number theory. To do this, he formed a ring called the "adeles".
Indeed, this is what my whole discussion so far has been leading up to! Adeles are a really nice formalism, and you pretty much need to understand them to follow what people are doing in work on th…