Timeline for Where am I going wrong in this interpretation of 1-dimensional geometric class field theory?
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| Sep 6, 2023 at 14:18 | vote | accept | oggledog | ||
| Sep 6, 2023 at 14:01 | answer | added | Will Sawin | timeline score: 5 | |
| Sep 6, 2023 at 10:03 | comment | added | Jason Starr | I thought that the OP was computing the Galois group of the field $\mathbb{F}_p(T)$, which is not topologically cyclic. Now I see that the OP is computing the Galois group of $\mathbb{F}_p$, which is topologically cyclic. | |
| Sep 6, 2023 at 2:38 | comment | added | Michael Barz | @JasonStarr Maybe I am mistaken -- but in this case, surely the Galois group is $\mathbb{Z},$ by comparing to the Picard group. It is true that every closed point gives a topological generator -- but that's because all closed points determine the same generator (in the double quotient, the Frobeniuses are all the same). | |
| Sep 6, 2023 at 0:48 | comment | added | Jason Starr | The Galois group has a topological generator for every closed point of $\mathbb{P}^1$. It is not topologically cyclic. | |
| Sep 5, 2023 at 16:10 | comment | added | R. van Dobben de Bruyn | (I'm just realising Neukirch only treats number fields, and there are always some difficulties passing to function fields. But I think that any reference should have some statement like this, where in the end you only really say something about the profinite completion of the locally profinite idèle class group $C_K$.) | |
| Sep 5, 2023 at 15:58 | comment | added | R. van Dobben de Bruyn | Hmm, if you look at the version L. Lafforgue proved (in Chtoucas de Drinfeld et correspondance de Langlands), there is always the running assumption that the representation is irreducible and its top exterior power has finite order. The final assumption removes the difference between $\mathbf Z$ and $\hat{\mathbf Z}$. In the abelian setting, this is also the version you find in Neukirch's Class field theory (the Bonn lectures, not his other book with the same name), Chapter III, Existence Theorem 7.8. | |
| Sep 5, 2023 at 15:28 | comment | added | oggledog | @R.vanDobbendeBruyn So is the "main theorem" that I stated false? Where could I find a correct statement? (Or do you happen to know the correct statement?) | |
| Sep 5, 2023 at 15:26 | comment | added | R. van Dobben de Bruyn | This difference between $\mathbf Z$ and $\hat{\mathbf Z}$ is supposed to happen, and this is why class field theory/the Langlands programme uses the (locally profinite) Weil group instead of the (profinite) Galois group. See for instance this question and its answers for some motivation. | |
| Sep 5, 2023 at 13:56 | history | edited | oggledog | CC BY-SA 4.0 |
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| Sep 5, 2023 at 13:55 | history | edited | oggledog | CC BY-SA 4.0 |
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| Sep 5, 2023 at 13:54 | history | edited | oggledog | CC BY-SA 4.0 |
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| S Sep 5, 2023 at 13:52 | review | First questions | |||
| Sep 5, 2023 at 17:53 | |||||
| S Sep 5, 2023 at 13:52 | history | asked | oggledog | CC BY-SA 4.0 |