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We have a natural surjective group homomorphism:

$\phi : SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/(n\mathbb{Z}))$

from which, given any subgroup $H<SL_2(\mathbb{Z}/(n\mathbb{Z}))$, we may take the pre-image under $\phi$. This gives us a "congruence subgroup". Most notably, we have

$\Gamma(n) : = \left \{\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \equiv \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] \mod n \right \} = \operatorname{Ker}(\phi)$

$\Gamma_1(n) : = \left \{ \left[ \begin{array}{cc} a& b \\ c & d \end{array} \right] \equiv \left[ \begin{array}{cc} 1 & * \\ 0 & 1 \end{array} \right] \mod n \right \} = \phi^{-1} ( U)$, U-unipotent matrices in $SL_2(\mathbb{Z}/(n\mathbb{Z}))$

$\Gamma_0(n) : = \left \{\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \equiv \left[ \begin{array}{cc} * & * \\ 0 & * \end{array} \right]\mod n \right \} = \phi^{-1}(T), $ T-triangular matrices in $SL_2(\mathbb{Z}/(n\mathbb{Z}))$

So I see these congruence subgroups everywhere but I never have found an explanation for any qualitative differences between modular forms of these different levels (or any congruence subgroups).

For instance,

  • If $f \in S_k(\Gamma_0(n))$, versus $f \in S_k(\Gamma_1(n))$, does this say anything about it's corresponding invaraints? (L-function, automorphic representations, Galois representations, jacobians, etc).

  • Is there any difference between the modular curves $X_{\Gamma}: = \mathbf{H} \cup P^1(\mathbb{Q})/\Gamma$, for these different levels?

  • The modularity theorem, for instance, guarantees a modular form of level $\Gamma_0(n)$. I have to imagine that $\Gamma_0$ was an important part of this implying Fermat's Last Theorem, but I could be wrong.

I'm interested in any sources available to help me understand any, if at all, differences between these subgroups and the rest of the story of modular forms.

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  • $\begingroup$ A newform $f \in \mathcal{S}_k(\Gamma_0(N))$ corresponds to a cuspidal automorphic representation of conductor $N$ and trivial central character. The automorphic representation corresponding to a newform $f \in \mathcal{S}_k(\Gamma_1(N))$, on the other hand, can basically have any central character whose conductor divides $N$ (modulo a parity condition on $k$). Classically, one can see this through the isomorphism $\mathcal{S}_k(\Gamma_1(N)) = \bigoplus_{\chi \pmod{N}} \mathcal{S}_k(\Gamma_0(N),\chi)$. Then note that $\mathcal{S}_k(\Gamma_0(N)) = \mathcal{S}_k(\Gamma_0(N),\chi_0)$,... $\endgroup$ – Peter Humphries Jan 25 '15 at 22:54
  • $\begingroup$ ...where $\chi_0$ is the principal character modulo $N$. So you should think of elements of $\mathcal{S}_k(\Gamma_0(N))$ as being particularly simple modular forms, namely the subset of $\mathcal{S}_k(\Gamma_1(N))$ consisting of cusp forms with principal nebentypus $\chi = \chi_0$. In practice, they can be easier to analyse because of this. $\endgroup$ – Peter Humphries Jan 25 '15 at 22:56
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There are inclusions $\Gamma(N) \subseteq \Gamma_1(N) \subseteq \Gamma_0(N)$, and correspondingly there is a tower of curves

$$X(N) \to X_1(N) \to X_0(N).$$

These are Galois coverings with Galois groups $\mathbb Z/N\mathbb Z$ and $(\mathbb Z/N\mathbb Z)^\times$ respectively (this is where the action of diamond operators comes from). The higher you are in the tower, the more modular forms there are.

There are various reasons why one might want to work with one level structure rather than another. For instance, $X_1(N)$ and $X(N)$ are sometimes easier to work with than $X_0(N)$ because they are (compactifications of) fine moduli spaces, whereas $X_0(N)$ is only a coarse moduli space. Having the universal property is quite convenient. The downside is having to work with larger spaces of modular forms; if you're interested in modular forms for $\Gamma_0(N)$ you then have to use the subspace which is invariant under Galois. But there's usually no loss of generality in working in a bigger space.

If you are working in finite characteristic, some level structures are much better than others. For instance, an elliptic curve over a finite field has $p$-torsion which is either $0$ or cyclic of order $p$, so such an elliptic curve doesn't admit a full level $p$ structure at all. However, it still admits a level $\Gamma_1(p)$ structure if it is ordinary.

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