Timeline for Recent developments on the Artin conjecture
Current License: CC BY-SA 2.5
3 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2011 at 2:26 | comment | added | David Hansen | ... This was carried out in a series of papers: Shepherd-Barron and Taylor proved the mod-$2$ modularity statement by a variant of Wiles's $3-5$ switch, first switching to the prime $5$ and then to the prime $3$; Dickinson proved the (very difficult) $2$-adic modularity lifting theorem by careful application of Wiles's techniques; and Buzzard and Taylor proved that unramified-at-$l$ $l$-adic lifts of modular mod-$l$ things are modular of weight one by gluing together overconvergent eigenforms defined on overlapping subsets of a rigid-analytic model of some modular curve. A tour de force! | |
| Mar 26, 2011 at 2:19 | comment | added | David Hansen | I think it is worth mentioning Taylor's largely successful program, which has since been supplanted by Khare-Wintenberger. Taylor's idea was to view an odd icosahedral representation $\rho$ as a $2$-adic Galois representation, whose mod-$2$ image matches the projective image of $\rho$ using the coincidence $A_5 \simeq SL_2(F_4)$. Then you can try to prove that mod-$2$ representations are modular, and then that $2$-adic lifts thereof which are unramified at $2$ are modular and correspond to weight one modular forms (all with some technical hypothesis, of course).... | |
| Mar 25, 2011 at 23:24 | history | answered | B R | CC BY-SA 2.5 |