Timeline for Langlands in dimension 2: the Yoshida conjecture
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2010 at 15:43 | comment | added | David Hansen | Because if it lands in a subgroup or has reducible semisimplification, its modularity can (I assume) be treated by Serre's conjecture over GL2 plus some instances of functoriality. Do people actually know how to prove analogues of Serre's conjecture for, say, the representation on $p$-torsion of an irreducible abelian surface defined $/ \mathbb{Q}$ with no rational $p$-torsion, at a prime $p$ of good ordinary reduction? | |
| Oct 6, 2010 at 12:18 | comment | added | TSG | Why the insistence on full image? I doubt anything nearly as strong is needed to apply R=T theorems, for example. | |
| Oct 6, 2010 at 4:36 | comment | added | David Hansen | I just realized the answer is likely yes - just take a Siegel modular form which isn't CAP or Yoshida, and look at the mod-$p$ reduction of the $p$-adic Galois representation attached to it by Laumon/Weissauer/Sorenson, for $p$ "generic". But does this obviously surject onto a $GSp_4(\mathbb{F}_{p^n})$? | |
| Oct 6, 2010 at 4:27 | comment | added | David Hansen | This is great, but doesn't it beg the question of proving modularity of the residual representation? Are there any known instances of modularity for a continuous odd representation $\rho: G_{\mathbb{Q}} \to GSp_4(\mathbb{F}_p)$ with full image? | |
| Oct 6, 2010 at 3:35 | comment | added | Emerton | The $p$-adic valuation of the $U_p$-eigenvalue of the overconvergent $p$-adic eigenform of weight $k$ should be $< k-1$. | |
| Oct 6, 2010 at 0:27 | comment | added | David Hansen | What is Coleman's criterion for the simpler GL2 case? I ask as a novice in p-adic modular forms. :) | |
| Oct 5, 2010 at 20:58 | history | answered | Olivier | CC BY-SA 2.5 |