4
$\begingroup$

I'm wondering where I can find a good reference about what is known about modular forms (especially cuspidal eigenforms) of full principal level $\Gamma(N)$, in terms of their Hecke theory, old/newform theory, rationality properties, action of automorphisms from $\text{GL}_2(\mathbb{Z}/N\mathbb{Z})$, functional equations, etc. Everywhere I look writers seem to address mostly only the nice multiplicity one cases of $\Gamma_0$ and $\Gamma_1$.

If possible, I'd like it in classical language, though automorphic language is fine too.

Improve this question
$\endgroup$
8
  • 5
    $\begingroup$ 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 $\Gamma_1(N)$. $\endgroup$
    – Peter Humphries
    Commented Mar 30, 2021 at 17:35
  • 3
    $\begingroup$ 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 $\endgroup$
    – François Brunault
    Commented Mar 30, 2021 at 18:17
  • 1
    $\begingroup$ 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 $\endgroup$
    – François Brunault
    Commented Mar 30, 2021 at 18:38
  • 1
    $\begingroup$ Also see LMFDB lmfdb.org $\endgroup$
    – Gerald Edgar
    Commented Mar 30, 2021 at 18:40
  • 1
    $\begingroup$ 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.) $\endgroup$
    – Kimball
    Commented Mar 30, 2021 at 19:25

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .