So I'm still a little new to Modular Forms. I have started reading Lang, I have watched Keith Conrad's lectures and I have read his notes.
I started to ask myself some questions. Let $\Gamma$ and $\Gamma'$ be subgroups of $\text{SL}_2(\mathbb{Z})$, both of finite index in $\text{SL}_2(\mathbb{Z})$, and let $M_k(\Gamma)$ be the space of Modular Forms of weight $k$ over $\Gamma$.
- If $\Gamma'$ is a subgroup of $\Gamma$ (whose index is denoted as $[\Gamma:\Gamma']$ — is it a standard notation ?), then $M_k(\Gamma)$ is a subspace of $M_k(\Gamma')$. Is there a way to relate $\dim(M_k(\Gamma)/M_k(\Gamma'))$ and $[\Gamma:\Gamma']$ ?
- Is there some sort of functoriality hidden ? What I mean is that, given a homomorphism $\varphi:\Gamma'\to\Gamma$, is there a way to construct a linear map $M_k(\varphi):M_k(\Gamma)\to M_k(\Gamma')$ ?
- If the latter happens to be true, how to relate this to (co-)homology ? Given a topological space $X$, one has functors $H_k$ that somewhat behave similarly as the $M_k$, and one may consider the graded ring $H_\star(X)=\displaystyle\bigoplus_{k}H_k(X)$, similarly to $M(\Gamma)=\displaystyle\bigoplus_{k}M_k(\Gamma)$. Are there connections ? More precisely, there are some geometric interpretations for the homology. Do those reflect similar interpretations for the group ? And how do some of the homological properties (like $H_n(X)=0$ if $n>\dim(X)$ for manifolds) translate in terms of the $M_k(\Gamma)$ ? Conversely, is there some way to relate some modular properties (such that $M(\text{SL}_2(\mathbb{Z})=\mathbb{C}[E_4,E_6]$, and analogues — when know —, to $M(\Gamma)$) with homology ?