Timeline for What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Dec 16, 2015 at 14:32 | comment | added | David Loeffler | @jdh The scanned typescript of Kato's article missing 20 pages, with "Prof. Greenberg" written on the cover in biro, seems to be all over the web; but I have a better one. Email me and I can send you a copy. | |
| Dec 16, 2015 at 13:45 | comment | added | jvo | @jdh: Too bad! I would offer you a full copy myself if I had one. (I am currently waiting to get full library access at my new university). Also,100 km is quite a trek (!) -- though perhaps only the beginning of it if you do go through Kato's entire argument … | |
| Dec 16, 2015 at 11:15 | comment | added | The Thin Whistler | @jvo: I had found the digital version you mentioned earlier, but it lacks pages 147-165. These pages however contain Thm. 14.4. It seems there is no choice for me but to travel the 100 kilometers. | |
| Dec 16, 2015 at 10:37 | comment | added | jvo | @jdh: Thanks for checking this out! There is a copy here at the moment: webusers.imj-prg.fr/~riccardo.brasca/pages/files/K2.pdf. I seem to recall that the deduction requires arguments from Kato's "Euler Systems, Iwasawa Theory, and Selmer Groups" too. Colmez and Scholl have also written expository accounts of this construction/argument: See Colmez, "La conjecture de Birch et Swinnerton-Dyer p-adique" and Scholl, "An introduction to Kato's Euler systems". | |
| Dec 16, 2015 at 9:31 | comment | added | The Thin Whistler | @jvo: The reduction type is crucial to Darmon's and Tian's argument. Does anyone of you know how to acquire a (scanned) electronic copy of Kato's article? I am about 100 km away from the next library (vacationing at my parents'), and the Société Mathématique de France does not allow their subscribing libraries to publish Astérisque digitally. | |
| Dec 15, 2015 at 22:57 | comment | added | jvo | Sorry, my bad: I had forgotten how Kato works, and should have looked more carefully before posting ... The results of Darmon-Tian are all stated for good ordinary reduction, though it is not clear to me whether this is strictly necessary for their arguments. These results for Kummer towers are fragmentary in any case, though I imagine more could be done in this direction now. | |
| Dec 15, 2015 at 20:56 | comment | added | David Loeffler | I'm unsure what's meant by the assertion "there is no suitable Euler system for the additive case". Kato's Euler system construction works, and combined with Rohrlich's work it solves the problem over the cyclotomic tower, for all quotienst of modular Jacobians and and all primes $p$ -- it's completely blind to the reduction type, to borrow @jvo's phrase. The existence of the p-adic $L$-function is more restricted but it isn't relevant for this argument. (I have no idea whether one needs more input from the cyclotomic theory to say anything over the larger Kummer tower, though.) | |
| Dec 15, 2015 at 16:39 | comment | added | jvo | Well … be bold, and mighty forces will come to your aid :) | |
| Dec 15, 2015 at 16:30 | comment | added | The Thin Whistler | So here I stand - poor fool! - once more, \\ no wiser than I was before. \\ (Goethe) | |
| Dec 15, 2015 at 14:32 | comment | added | jvo | Correct, at least as far as I know. The nonvanishing theorems are blind to the reduction type, but the Euler systems and p-adic L-functions constructions (which are typically only available for good ordinary, multiplicative, or special cases of good supersingular reduction) depend greatly upon it. | |
| Dec 15, 2015 at 14:08 | comment | added | The Thin Whistler | Yes. But to my knowledge, there is no suitable Euler system for the additive case. Therefore, not much is yet known in the additive case, am I right? | |
| Dec 15, 2015 at 10:53 | vote | accept | The Thin Whistler | ||
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| Dec 15, 2015 at 10:47 | vote | accept | The Thin Whistler | ||
| Dec 15, 2015 at 10:53 | |||||
| Dec 15, 2015 at 10:43 | vote | accept | The Thin Whistler | ||
| Dec 15, 2015 at 10:43 | |||||
| Dec 15, 2015 at 10:37 | history | answered | jvo | CC BY-SA 3.0 |