Timeline for Explaining the number field-function field analogy
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Jul 20, 2014 at 20:33 | comment | added | Urs Schreiber | There is now some related material collected on the nLab here: ncatlab.org/nlab/show/function+field+analogy | |
| Oct 11, 2010 at 14:34 | comment | added | BCnrd | 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). | |
| Oct 11, 2010 at 11:29 | history | edited | Charles Matthews | CC BY-SA 2.5 |
downcase
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| Oct 11, 2010 at 11:27 | answer | added | Charles Matthews | timeline score: 8 | |
| Oct 11, 2010 at 10:52 | comment | added | Cam McLeman | 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." | |
| Oct 11, 2010 at 10:29 | history | asked | anonymous | CC BY-SA 2.5 |