Question

Let ${\mathbb T}$ denote the ordinary $\Lambda$-adic Hecke algebra of say tame level $N$. If I specialize ${\mathbb T}$ to a classical weight $k \geq 2$, then it is proven by Hida that the result is the Hecke algebra of classical $p$-ordinary forms of weight $k$ and level $pN$. (Here by specialize I mean mod out by the prime ideal $p_k \subseteq \Lambda = {\mathbb Z}_p[[\Gamma]]$ generated by $[\gamma] - \gamma^{k-2}$ where $\gamma$ is a topogical generator of $\Gamma$.) I guess this is Hida's control theorem.

My question is the following: if one takes $\kappa$ to be some $p$-adic weight (but not necessarily classical), how can one describe ${\mathbb T}/ p_{\kappa} {\mathbb T}$?

I assume the answer is that the result is the Hecke algebra of the space of overconvergent modular forms of weight $\kappa$ and slope 0 -- but I don't have a proof of this nor a reference. Either would be greatly appreciated!

Answer 1

I don't know if this is always true, and I don't think this is known in all cases, even after inverting $p$.

What is known and what is more or less formal to prove (it depends of your starting point, and in particular, of how you define the "$\Lambda$-adic Hecke algebra of tame level $N$) is that there is a natural surjection, whose kernel is nilpotent, from ${\mathbb T}/p_\kappa {\mathbb T}$ to the Hecke algebra of the space of overconvergent modular forms of weight $\kappa$ and slope 0. But then, the question of whether the kernel is trivial is difficult to settle without knowing more on the local structure of $\mathbb T$ near the points above $\kappa$.

Edit to answer HP's comment below : Let us consider a "toy model": imagine your base ring is $\mathbb{Z}_p$, your module of ordinary p-adic form is $M=\mathbb{Z}_p^2$, and there is only one Hecke operator around, namely $T$ acting on the canonical basis of $M$ by $T(e_1)=p e_1$, $T(e_2)=0$. Then $\mathbb{T}=\mathbb{Z}_p[T]/T(T-p)$ is free of rank $2$ over $\mathbb{Z}_p$, but on $M/pM$, $T$ acts like $0$ et the Hecke algebra on $M/pM$ is just $\mathbb{F}_p$ which is not $\mathbb{T}/p$.

Now what I say is not that this kind of examples actually happen on the eigencurve, but that without supplementary hypothesis (like level $1$, artfully restricting to new forms, etc.) no one, as far as I know, can rule this out.

Answer 2

I guess nothing new occurs. First of all, we know the algebraic strucutre of $\Lambda$ and we know that its non-maximal ideals are all primcipal, generated by element of the form $$ p^\mu\prod_if_i(T)^{a_i} $$ for some $\mu\geq 0$ and irreducible distinguished polynomials $f_i$. In particular, you can write $(1+T)^\kappa-1$ for $\kappa\in\mathbb{Z}_p\setminus\mathbb{Z}$ in the above form, so falling back to some classical specialization.

If you prefer a better explanation, a result of Coleman tells you that all overconvergent $p$-adic modular forms of slope $< k-1$ are classical (this is Theorem 6.1 of Coleman's "Classical and overconvergent modular forms", Inventiones 1996)