Timeline for Local components of quaternionic modular forms
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 20, 2013 at 13:54 | comment | added | Emerton | Dear Konrad, If $\overline{\rho}$ is a mod $p$ Galois rep'n coming from level prime to $q$, then there are some conditions (level-raising conditions, due to Ribet) which must be satisfied if $\overline{\rho}$ can also arise from a modular form which is Steinberg at $q$. When you choose the TW primes, you choose them in such a way that these level-raising conditions are not satisfied. In this way, the possibility of Steinberg at the TW primes is ruled out. (Look in any discussion of the TW method, e.g. the one in the Fermat's Last Theorem book, and you will see this explained.) Regards, | |
| Jun 20, 2013 at 10:12 | comment | added | Konrad | I meant "Steinberg-ness" in my last comment... | |
| Jun 20, 2013 at 9:02 | comment | added | Konrad | Thank you both. It seems I should first thoroughly read up on JL. Still I don't see, how the added level at TW primes rules out the possibility of Eisenstein-ness. | |
| Jun 19, 2013 at 19:45 | comment | added | Emerton | ... Jacquet--Langlands? One of the basic properties of JL is that its image consists of those GL_2 reps. that are discrete series, i.e. twist of Steinberg or supercuspidal.) The short answer is that your comment about unramified or Steinberg is completely wrong. Regards, | |
| Jun 19, 2013 at 19:44 | comment | added | Emerton | Dear Konrad, Remember that at all but finitely many primes, $D^{\times}$ is just $\GL_2$, so the local components can be whatever they want to be. In particular, the TW primes will always be primes where $D$ is split, and then we add level structure that, from the automorphic viewpoint, amounts to considering automorphic reps. whose local comps. at the TW primes are tame principal series. At the ramified primes, the local components have to be reps. of the local quat. alg., which are ... whatever they are. (Perhaps you want to know what they correspond to on the GL_2 side of... | |
| Jun 19, 2013 at 17:33 | comment | added | Joël | Where did you read about the claims that the local components at $\nu$ are always Steinberg or unramified? That doesn't seem right. And by the way, in this statement, what is $\nu$? any finite place of $F$? any finite place where $D$ is split? | |
| Jun 19, 2013 at 16:50 | history | asked | Konrad | CC BY-SA 3.0 |