There is a general circle of ideas according to which true statements about number fields should have analogues in function fields. As best I can tell, the fact that this seems to work is pretty mysterious. The only results I know directly relating the two come from logic, such as Ax-Kochen, and these are limited to first-order statements in restricted languages. But the analogy apparently goes well beyond such statements. Are there any theorems/conjectures/observations that would explain why the analogy is a good one? I am looking for statements that allow one to go directly from the function field case to the number field case or vice versa.