# Explaining the number field-function field analogy

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.

• I think this question is a little too vague as stands. For example, a lot of the analogy turns out not to be an "analogy" at all, but rather the observation that rings of integers in both cases are Dedekind domains. Once that one fact is the established, a lot of the basic analogy falls into place. The abstract class field theory developed by Neukirch put class field theory into a similar position -- one only needs a mild cohomological lemma to be satisfied for both number fields and function fields, and then you get all of class field theory "for free." – Cam McLeman Oct 11 '10 at 10:52
• One striking case where there is direct logical connection is the work of Waldspurger that reduced the Fundamental Lemma in char. 0 to the case of positive characteristic. Here the "link" between these two worlds begins with the observation that sufficiently ramified extensions of $p$-adic integer rings "look like" sufficiently ramified extensions of valuation rings of local function fields when everything is reduced modulo $p$ (made precise in different ways by Deligne and Fontaine). – BCnrd Oct 11 '10 at 14:34
• There is now some related material collected on the nLab here: ncatlab.org/nlab/show/function+field+analogy – Urs Schreiber Jul 20 '14 at 20:33