This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler.
Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb Z$-subalgebra of endomorphisms of $S_2(\Gamma_1(N))$ generated by Hecke operators $T_\ell$ for all prime $\ell\notin\Sigma$. If $\ell\in\Sigma$ implies that $\ell\nmid N$, then it is well-known that the index of $\mathbb T^{\Sigma}$ in $\mathbb T^{\varnothing}$ is a finite power of 2. I first learned this result in
Wiles, Andrew Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3 Lemma page 491
where the given proof is attributed to F.Diamond and where the statement is credited to
Ribet, Ken Multiplicities of $p$-finite mod $p$ Galois representations in $J_{0}(Np)$. Proposition 2
Thanks to David's question quoted above, I have been wondering about the actual possible values for the index of $\mathbb T^{\Sigma}$ in $\mathbb T^{\Sigma'}$ when $\Sigma'\subset\Sigma$. Here then are my actual questions.
A concrete one:
Question 1 Do you know of actual examples of non-trivial index of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\varnothing}$ and of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\Sigma'}$ when $\Sigma'$ contains primes dividing $N$?
A more theoretical (but also vaguer) one:
Question 2 Is there something known about the power of 2 that can occur as index of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\varnothing}$? about the index of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\Sigma'}$?
Regarding the second half of the first question, A.Wiles is careful in pointing that the argument given in his aforementioned article does not prove that $\mathbb T^{\Sigma\cup\{\ell\}}$ is equal to $\mathbb T^{\{\ell\}}$ when $\ell|N$ but neither does he say that the result could be false.
In the comments of Omitting primes from a Hecke algebra Kevin Ventullo seems to suggest that one maybe able to retrieve the missing operators from the local Langlands correspondence (in its $p$-adic Hodge theoretic incarnation at $\ell=p$) but I have been unable to understand his precise idea, especially as the proof I quoted (which is the only one I know) involves representations with coefficients in Artinian algebras, so are not obviously compatible with Weil-Deligne representations.
