On size of Hecke algebras.

Question

Let $G$ be a subgroup in $SL_2(\mathbb{Z})$ and $S_k(G)$ be the space of cusp (automorphic?) forms invariant by any element of $G$ of weight $k$.

Question 1: Generally for two arithmetic subgroups $G < G'$ in $SL_2(\mathbb{Z})$, is there a surjection $\operatorname{End}(S_k(G)) \twoheadrightarrow \operatorname{End}(S_k(G'))$?

For example, what is the relation between $\operatorname{End}(S_k(\Gamma_0(N)))$ and $\operatorname{End}(S_k(\Gamma_1(N)))$?

Question 2: In Wiles's Theorem $R = T$, which is the right one between $T = \operatorname{End}(S_2(\Gamma_0(N)))$ or $T = \operatorname{End}(S_k(\Gamma_1(N)))$?

Pierre MATSUMI

Answer 1

For any space $S$ of modular forms one has a Hecke algebra $T$ associated to $S$, defined as the $\mathbf{Z}$-subalgebra of $\operatorname{End}_{\mathbf{C}} S$ generated by the Hecke operators $T_n$. This is much smaller than the matrix algebra $\operatorname{End}_{\mathbf{C}} S$ itself. In particular it's commutative (which is a prerequisite for an $R = T$ theorem, since universal deformation rings are commutative by definition).

If $\Gamma \le \Gamma'$ then $S_k(\Gamma') \subset S_k(\Gamma)$ (for any $k$) but that doesn't give any map between the corresponding Hecke algebras $T'$ and $T$ in general (because the Hecke operators $T_n$ of level $\Gamma$ and $\Gamma'$ aren't necessarily compatible). However, if $\Gamma$ and $\Gamma'$ are both congruence, of levels $N$ and $N'$ say, and $N$ and $N'$ have the same set of prime factors, then one gets a surjection $T \to T'$. For instance, this applies to $\Gamma_1(N)$ and $\Gamma_0(N)$ for any $N$.

In Wiles' original $R = T$ theorem, his $T$ was not quite a Hecke algebra in the above sense, but the completion of the Hecke algebra associated to $S_2(\Gamma_0(N))$ at a "non-Eisenstein maximal ideal".

Answer 2

Regarding your first question, I might know where your confusion is coming from. Some people also would refer to a Hecke algebra by the space of complex valued function $f$ on $SL_2(\mathbb{Q})$ or $GL_2^+(\mathbb{Q})$ biinvariant under $G$ respective $G'$ (the algebra multiplication defined via convolution), especially if you are coming from an adelic point of view and $G, G'$ are congruence. If $G$ is finite index in $G'$, there is a inclusion (by restriction) as well as a projection (by averaging over double cosets).

But the usual Hecke operators due to Hecke are really only a subalgebra of these algebras. But it is e.g. a commutative algebra if $G = SL_2(\mathbb{Z})$ or $G=GL_2^+(\mathbb{Z})$, because the $( GL_2(\mathbb{Q}_p), GL_2(\mathbb{Z}_p))$ is a Gelfand pair for each $p$.