Timeline for Characters and weights of Hilbert modular forms
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2020 at 5:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 18, 2019 at 4:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 18, 2019 at 8:15 | comment | added | Leray Jenkins | One thing I forgot to mention, but this is slightly tangential, is that usually weights for Hilbert modular forms also have a parity condition, i.e. you require all the $k_\sigma$ to have the same parity. This is to make sure your spaces of Hilbert modular forms aren't empty for silly reasons. In any case, I think you got the idea :) | |
| Sep 18, 2019 at 3:43 | answer | added | Jon Aycock | timeline score: 1 | |
| Sep 18, 2019 at 1:44 | comment | added | Jon Aycock | Right, we can clearly do this after base changing to $\mathbb{R}$. I guess we do get that $\chi(n)$ is in $\overline{\mathbb{Z}}$, the integral closure of $\mathbb{Z}$ in $\mathbb{R}$, so there is some integrality preserved. Is the character defined over $\overline{\mathbb{Z}} \subset \mathbb{R}$? If so, it shouldn't be too hard to have it be defined over $\mathcal{O}_F$ (since $F$ is assumed Galois over $\mathbb{Q}$). | |
| Sep 17, 2019 at 18:16 | comment | added | Leray Jenkins | I think what one does is, for $(k_\sigma)_\sigma \in \mathbb{Z}^\Sigma$, one sends $n$ to $\prod_\sigma \sigma(n)^{k_\sigma}$ where $\Sigma$ is the set of embedding of $F$ into $\mathbb{R}$. This clearly preserves the integrality. If you then take $k$ to be parallel you then see that this is just the $k$-th power of the norm. | |
| Sep 16, 2019 at 19:08 | history | asked | Jon Aycock | CC BY-SA 4.0 |