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.
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9$\begingroup$ 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." $\endgroup$– Cam McLemanCommented Oct 11, 2010 at 10:52
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2$\begingroup$ 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). $\endgroup$– BCnrdCommented Oct 11, 2010 at 14:34
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$\begingroup$ There is now some related material collected on the nLab here: ncatlab.org/nlab/show/function+field+analogy $\endgroup$– Urs SchreiberCommented Jul 20, 2014 at 20:33
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I think your statement could usefully be sharpened in a couple of ways.
Firstly, the state-of-the-art is that true statements for function fields are expected to have analogues for number fields. The outstanding example here is naturally the Riemann hypothesis. Where conjectures for number fields are present, the way of working is via the heuristic that the function field analogue may be sought, and then proved. This has been taken a long way for the Langlands philosophy, for example.
The other major point is that historically what came first was analogies between Riemann surface theory and number fields. Hilbert seems to have conjectured the main outlines of class field theory using complex curves and Jacobians as the source of inspiration. Certainly the prior Dedekind-Weber geometric view fed into that. Subsequently we get "global field" as a kind of middle term: curves over finite fields are a little closer to number fields than complex curves. The attitude of Weil's Basic Number Theory is to develop the parallelism to the point of a common vocabulary (which is now widely used). Global fields can be studied successfully using local compactness, would be a succinct summary. Pontryagin duality, for example, can take much of the strain, and this theory is of course of broader application than number theory.
Of course we don't yet know why this works.
answered Oct 11, 2010 at 11:27
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$\begingroup$ Weil's thoughts on analogies ams.org/notices/200503/fea-weil.pdf $\endgroup$ Commented Nov 18, 2019 at 14:33