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!



Non-classical specializations of Hida families
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. 


