Timeline for Level vs. conductor of a supercuspidal representation
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 18, 2020 at 19:40 | comment | added | Kimball | @LSpice Well, I didn't want to flesh things out anymore since there's and your answer was already accepted. Oh, I didn't realize by unramified you meant "unramified supercuspidal" in the sense that Bushnell, Henniart, etc use (which I personally dislike for what I hope are obvious reasons). Yes, then there might be some normalization of the level one needs to go between your answer and my comment. | |
| May 18, 2020 at 17:46 | comment | added | LSpice | @Kimball, definitely, and I think you should post it as an answer. However, I'm pretty sure that, even for supercuspidals, $e$ can be $1$ or $2$ (or else I'm totally misunderstanding their notation, which is entirely possible). | |
| May 18, 2020 at 16:30 | comment | added | Kimball | @LSpice Maybe I should have clarified, but the question was about supercuspidal $\pi$, and I am only stating what you get when $\pi$ is discrete series, so $e=2$ in your notation, using the normalization of level as in Bushnell-Henniart's book. I haven't looked at Bushnell-Henniart-Kutzko, at least not in detail, but I thought the references I use (which are just for GL(2)) might be a candidate for "a more elementary reference" that you mentioned in your answer. | |
| May 18, 2020 at 15:18 | comment | added | LSpice | @Kimball, it may be necessary to be cautious here: I think that there are notions of normalised and of un-normalised level. According to Bushnell, Henniart, and Kutzko - Local Rankin–Selberg convolutions for $\operatorname{GL}_n$, the level is an integer $m$, and $\frac m e = \frac1 2(f - 1)$, where $e = 1$ if $\pi$ is unramified and $e = 2$ if $\pi$ is unramified. Probably you are using a normalised notion, where the level is the rational number $m/e$? | |
| May 18, 2020 at 1:31 | comment | added | Kimball | I explained how to get this from standard references in Section 2.2 of my basis problem paper. You get that the level is one less than one half of the conductor. | |
| May 17, 2020 at 23:14 | vote | accept | user15243 | ||
| May 17, 2020 at 21:28 | answer | added | LSpice | timeline score: 2 | |
| May 17, 2020 at 21:12 | comment | added | LSpice | I don't know if it's just me, but the PDFs to which you linked won't load. I changed the arXiv PDF link to an abstract link, as is the usual convention, but I had to guess at the Breuil paper, which I think is Breuil and Mézard - Multiplicités modulaires et représentations de $\operatorname{GL}_2(\mathbb Z_p)$ et de $\operatorname{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$ en $\ell = p$ (MSN). | |
| May 17, 2020 at 21:10 | history | edited | LSpice | CC BY-SA 4.0 |
Names of papers
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| May 17, 2020 at 20:42 | history | asked | user15243 | CC BY-SA 4.0 |