Timeline for Symmetric powers of Ramanujan tau-function
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3 events
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| Feb 26, 2018 at 4:42 | comment | added | just-a-guest | (cont.) There's also the 4th paragraph of pg 3 CT, which seems to suggest that the level-raising argument only works for $l=5,7$, whereas they actually skirt the above Arthur issue (though similar others remain), by saying that "Schur reps" of unitary type can be handled by deformation theory in $GL_9(\bar Z_7)$. Alternatively, in Math Reviews, Wiese suggests that the failure to handle 10 is more "random", in the sense one needs $k-1$ to be prime, noting that the main result is: $C(l-1)$ implies $C(l+1)$ for $l$ prime, where $C$ is the principal Conjecture in the realm, provable for $l=5,7$. | |
| Feb 25, 2018 at 23:04 | comment | added | just-a-guest | Perhaps it is useful to say briefly why 8 is the current limit. E.g., in the Shahidi method, they ran out of the exceptional Lie groups ($E_8$ provided the 4th symmetric power). Here, I am no expert, but I think the limitation is the first sentence of the last paragraph on page 2 of CT: namely that you have a decomposition into essentially orthogonal reps of degrees 4 and 5, which has been considered/classified by Arthur/Mok/(Moeglin-Waldspurger?), and the analogous does not exist for higher powers? | |
| Feb 21, 2018 at 9:47 | history | answered | David Loeffler | CC BY-SA 3.0 |