Timeline for Modular forms on $\Gamma(N)$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 31, 2021 at 16:26 | comment | added | Peter Humphries | Yes, that's correct. | |
| Mar 31, 2021 at 14:24 | comment | added | xir | @PeterHumphries I managed to find it in Miyake, I guess you meant z/N and not z/q, right? | |
| Mar 31, 2021 at 14:23 | comment | added | xir | @FrançoisBrunault thanks, that's very helpful! | |
| Mar 30, 2021 at 19:25 | comment | added | Kimball | Did you try Cohen-Stromberg or Miyake? Those are fairly thorough treatments of modular forms, and they probably at least explain Peter's comment. (Diamond-Shurman probably explains Peter's comment as well.) | |
| Mar 30, 2021 at 18:40 | comment | added | Gerald Edgar | Also see LMFDB lmfdb.org | |
| Mar 30, 2021 at 18:38 | comment | added | François Brunault | In his PhD thesis, Weinstein has computed the decomposition of $S_k(\Gamma(N))$ as a $\mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z})$-module. math.bu.edu/people/jsweinst/jswthesis.pdf | |
| Mar 30, 2021 at 18:17 | comment | added | François Brunault | If you adopt the automorphic language, then by Casselman's theorem, for any irreducible smooth representation of $\mathrm{GL}_2(\mathbb{Q}_p)$, the new vector will be on $\Gamma_1(p^r)$ for some $r$. See arxiv.org/abs/1008.2796 | |
| Mar 30, 2021 at 18:04 | history | edited | xir | CC BY-SA 4.0 |
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| Mar 30, 2021 at 17:35 | comment | added | Peter Humphries | I don't think this kind of thing is written down anywhere. The key point is that $$\mathcal{M}_k(\Gamma(N)) = \bigoplus_{\chi \pmod{N}} \mathcal{M}_k(\Gamma_0(N^2),\chi).$$ More precisely, given $f \in \mathcal{M}_k(\Gamma_0(N^2),\chi)$, the function $g(z) = f(z/q)$ is an element of $\mathcal{M}_k(\Gamma(N))$, and every element arises in this way (up to taking appropriate linear combinations). So all of the key properties of $\Gamma(N)$ can be reduced to properties for $\Gamma_0(N)$ and $\Gamma_1(N)$. | |
| Mar 30, 2021 at 17:22 | history | asked | xir | CC BY-SA 4.0 |