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.

  • $\begingroup$ Did you calculate any examples? It shouldn't be hard to do so (e.g. in Sage). $\endgroup$ Jul 5 '13 at 11:47
  • $\begingroup$ @David Loeffler. I'm not proud of it, but I admit that the amount of work I put in trying to answer my own questions is very low: my curiosity was piqued by your previous question and I wondered if someone on MO would know more, and that's about it. $\endgroup$
    – Olivier
    Jul 5 '13 at 20:26

Examples of the phenomenon alluded to in Question 1 are actually plentiful. The first that came to my attention is described in

M.Emerton $p$-adic families of modular forms (after Hida, Coleman, and Mazur). Séminaire Bourbaki. Vol. 2009/2010. Exposés 1012–1026. Astérisque No. 339 (2011).

I reproduce it here.

Consider $\mathbb T(23)$ and $\mathbb T^{\{2\}}(23)$ (both acting on $S_2(\Gamma_{1}(23)$). The first ring is isomorphic to $\mathbb Z[(1+\sqrt{5})/2]$ whereas the second is sent to $\mathbb Z[(1+\sqrt{5})]$ through this isomorphism and is thus of index 2.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.