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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!

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Lemma 1 of Buzzard-Taylor "Companion forms and weight 1 forms" confirms your expected answer for integral weights. Perhaps one can adapt the proof to work for general weight weight $\kappa$? –  jnewton Aug 4 '12 at 22:17
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2 Answers 2

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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.

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I have tried to answer to you. Not knowing your background, I am not sure that it will make much sense. Please ask more if needed. –  Joël Aug 23 '12 at 20:09
    
Hi Joël, I'm having trouble with the toy model. If the Hecke eigenvalues of the forms $e_1$ and $e_2$ agree modulo $p$, then the forms themselves must agree modulo $p$. But then $(e_1-e_2)/p$ has integral coefficients, so should also belong to the module $M$. –  Kevin Ventullo Aug 24 '12 at 0:47
    
Ah, I see the problem now. In the Hida context, the analogue of $p$ need not be a principal ideal. –  Kevin Ventullo Aug 24 '12 at 1:52
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Kevin, if you work in tame level N>1, and do not use the operators at primes dividing N in defining the Hecke algebra, then the equality of Hecke eigenvalues does not imply the equality of the forms. –  Joël Aug 24 '12 at 12:42
    
@Joel: Many thanks! This clears up all of my confusions. –  H P Aug 28 '12 at 21:59
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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)

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I'm not sure I understand these remarks - what do you mean by "falling back to some classical specialization"? The question is asking whether the specialisation of a Hida family at a non-classical weight is overconvergent, so I don't see the relevance of Coleman's result (which tells us about overconvergent forms of classical weight). –  jnewton Aug 15 '12 at 9:09
    
Oh, yes, you are perfectly right. I misread your question, apologies. –  Filippo Alberto Edoardo Aug 18 '12 at 22:38
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