Timeline for Connection between isomorphisms of algebraic topology and class field theory
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 3, 2016 at 4:12 | vote | accept | Julian Rosen | ||
| Nov 27, 2010 at 17:28 | answer | added | Ben Wieland | timeline score: 12 | |
| Nov 26, 2010 at 14:42 | comment | added | BCnrd | Dear Alex: Minhyong's essay is largely focused on the saga of base points, which is not so relevant in the abelian case, and although he mentions Weil's awareness of the abelian analogy, he doesn't get into the perspective of Jacobians that is needed to make it precise. It seems to me that the story of Jacobians and Picard functors (which is somewhat different from that of etale fundamental groups) is at the heart of unifying the two situations Julian asks about. That's the sense in which Minhyong's essay seems to be about something else (though it is still important information!). | |
| Nov 26, 2010 at 9:44 | comment | added | Alex B. | Dear Brian, maybe I am reading too much into Minhyong's answer, but I think he does mention the connection between the fundamental group of a variety and the Galois group of the maximal unramified extension of the function field. He also mentions that "Weil was fully aware that homology and class groups are somehow the same". The rigorous connection to the number field case, given by étale fundamental groups, is explained in Szamuely's book that Davidac897 links to, and particularly in the 2 extra chapters, linked to by AS. Or am I missing something and that's not what this question is about? | |
| Nov 26, 2010 at 8:54 | comment | added | BCnrd | Dear Julian: the above link doesn't address abelian unramified case. There are really two theorems, and one bridge. Compact Riemann surfaces and global fn fields are subsumed by "geometric class field theory" of Lang-Rosenlicht (using Jacobians to classify finite abelian coverings); see Serre's book. Global fn fields and number fields are subsumed by the usual CFT (together with Kummer sequence to relate cohomology of $\mathbf{G}_m$ to that of $\mu_n$, and the link between Jacobian and Pic). So global fn fields are the bridge. Weil wrote to his sister about it. Talk to K.P. or M.Z. | |
| Nov 26, 2010 at 8:21 | history | edited | BCnrd | CC BY-SA 2.5 |
Fixed up the description of Pic(O_K) with correct coefficients; (O_K)* was wrong.
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| Nov 26, 2010 at 7:21 | comment | added | Alex B. | mathoverflow.net/questions/546/… | |
| Nov 26, 2010 at 7:11 | history | asked | Julian Rosen | CC BY-SA 2.5 |