Timeline for Hilbert Modular Newforms
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Oct 22, 2010 at 15:29 | comment | added | user1073 | Olivier - I believe that you are looking at Ogg's Corollary (on page 104) and not his Corollary 1 (on page 107). The Corollary that you refer to assumes that the form have trivial character. | |
| Oct 22, 2010 at 15:25 | vote | accept | CommunityBot | moved from User.Id=1073 by developer User.Id=69903 | |
| Oct 22, 2010 at 15:18 | comment | added | Kevin Buzzard | Note that Theorem 4.6.17 of Miyake's book on modular forms is a reference for the statement over the rationals. I wonder whether vytas' answer can somehow be translated into a similar "low-level" proof of the assertion in the Hilbert case. | |
| Oct 22, 2010 at 10:00 | answer | added | vytas | timeline score: 5 | |
| Oct 22, 2010 at 9:34 | comment | added | Olivier | Ben-Ogg's Cor. 1 says that $a(p)=0$ when $e_{p}>1$ and I couldn't see how this was excluded by W.Li (I still can't in fact, but that's my problem, isn't it?). | |
| Oct 22, 2010 at 6:38 | answer | added | Kevin Buzzard | timeline score: 11 | |
| Oct 22, 2010 at 6:06 | comment | added | Kevin Buzzard | @Olivier: I don't think $\pi$ can be supercuspidal. The condition that the form has conductor $q^t$ at $q$ (here $q$ is a prime ideal and $t\geq1$) and the form also has conductor $q^t$ at $q$ implies that $\pi$ is ramified principal series at $q$, with one unramified and one ramified character, so the result is true. | |
| Oct 22, 2010 at 5:45 | comment | added | user1073 | Olivier - Which of the hypotheses in Ogg's Cor. 1 do you not see as being verified? | |
| Oct 22, 2010 at 5:35 | comment | added | Olivier | Like unknown above, I am slightly perplexed by the assertions. When $\pi(f)$ is supercuspidal at $v$, I don't see how to prove this result. In fact, I am already rather surprised by the proof in W.Li' article. The results you quote is proved on page 295 but the proof uses a corollary of Ogg. However, I can't see how the hypotheses of the corollary of Ogg are verified. If you are happy to assume that $\pi(f)_{v}$ is not supercuspidal, then this should follow from the description of the Langlands $L$-factor at $v$. | |
| Oct 22, 2010 at 0:13 | comment | added | unknown | You can read these things in a paper of Casselman in the second of Antwerp volumes "Modular functions of one variable" (pp 119-120 might be useful). I think that a theorem of the type $a_p\neq 0$ can be proved only if one knows a priori that $\pi_p$ is not supercuspidal. | |
| Oct 22, 2010 at 0:07 | comment | added | unknown | In representation theoretic terms if $f$ is a classical newform of conductor $N$ and $\Pi$ is the associated automorphic representation of $GL_2$, then the Hecke eigenvalue $a_p$ is closely related to the local component $\Pi_p$ at $p$ of $\Pi$. If $p$ does not divide $N$, then $\Pi_p$ is obtained by inducing two characters from the Borel that can be determined from $a_p$. If $p$ does divide $N$ then the picture is (I think) more complicated. However if $\Pi_p$ is a twist by a character $\chi$ of the special representation, then the eigenvalue $a_p$ is related to $\chi$ by a simple formula. | |
| Oct 21, 2010 at 21:02 | history | asked | user1073 | CC BY-SA 2.5 |