Timeline for Kummer generator for the Ribet extension
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Sep 25, 2010 at 9:38 | comment | added | Chandan Singh Dalawat | Good to see that you are still mulling over it. | |
| Sep 21, 2010 at 18:41 | comment | added | Franz Lemmermeyer | There are two cases in Washington's proof; in the second case he constructs a unit that gives a Kummer generator of a suitable unramified construction, but in the first case he does not. Without having gone through the details I bet that if Vandiver holds, we are always in the second case, and that the problematic case is the first case in Washington's proof. | |
| Sep 21, 2010 at 14:28 | vote | accept | Chandan Singh Dalawat | ||
| Sep 9, 2010 at 10:55 | comment | added | Franz Lemmermeyer | The problem, as far as I can see, is his comment "Such a thing exists". I cannot see that. | |
| Sep 9, 2010 at 4:33 | history | edited | Chandan Singh Dalawat | CC BY-SA 2.5 |
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| Sep 9, 2010 at 3:16 | comment | added | Chandan Singh Dalawat | @Franz : Great answer ! But Cam (and Washington 15.8) seem to be constructing $E$ by adjoining the $p$-th roots of some (not very explicit) unit of $K$, irrespective of whether $p$ satisfies Vandiver or not. You seem to be skeptical that this can be done. What is going on ? | |
| Sep 9, 2010 at 3:01 | history | edited | Chandan Singh Dalawat | CC BY-SA 2.5 |
Changed \ell-1 to p-1 in the summation
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| Sep 8, 2010 at 13:30 | comment | added | Cam McLeman | Agreed! Link much appreciated. | |
| Sep 8, 2010 at 12:56 | history | edited | Franz Lemmermeyer | CC BY-SA 2.5 |
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| Sep 8, 2010 at 10:05 | comment | added | Chandan Singh Dalawat | Thank you very much for putting your excellent notes (rzuser.uni-heidelberg.de/~hb3/publ/pcft.pdf) online. Future generations will owe you a debt of gratitude... (By the way, it is $l^2$, not $l^n$, in the title of Pollaczek's paper). | |
| Sep 7, 2010 at 10:28 | history | edited | Franz Lemmermeyer | CC BY-SA 2.5 |
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| Sep 7, 2010 at 9:10 | comment | added | Chandan Singh Dalawat | ``[T]here will be more than one if the irregularity index is $>1$''. Yes, but we are fixing the pair $p$, $k$ (such that $p|B_k$), and then one expects (Iwasawa's conjecture) there to be a unique line $D\subset K^\times/K^{\times p}$ such that $E=K(\root p\of D)$ is everywhere unramified over $K$ and such that $\Delta$ acts on ${\rm Gal}(E|K)$ via $\chi^{1−k}$. | |
| Sep 7, 2010 at 8:50 | comment | added | Chandan Singh Dalawat | Thanks for the reference to Pollaczek and for the remarks. Ribet's major contribution in this story is not so much the result as the method, which has been exremely fruitful. See for example the notes of Mazur talk at the Ribet conference (math.harvard.edu/~mazur). Waiting anxiously for your lost construction to be found and put online. | |
| Sep 7, 2010 at 6:41 | history | answered | Franz Lemmermeyer | CC BY-SA 2.5 |