# p-adic modular forms, Hecke algebra, deformation theory and modular curves.

Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation $\bar{\rho}$ which is $p$-distinguish and minimal and such that the restriction of $\bar{\rho}$ to $G_{\mathbb{Q}(\sqrt{(-1)^{(p-1)/2}p)}}$ is absolutely irreducible.

let $T$ be the localisation of $h^{ord}(N,\mathcal{O})$ by $\mathfrak{m}$.

In the literature I see that we assume often that the tame level is a square free (since any semi-stable elliptic curves has a square free conductor)

There is some modularity lifting theorems $R^{ord}=T$ without the assumption that the tame level of $T$ is a square free?

I don't see immediately when we use this condition in the Horizontal control (in the Hecke side)?

To give a reference: in the article by Böckle referenced below, he proves an $R^{\mathrm{ord}}=T$ type theorem (Thm. 3.9) without assuming that the tame level is square free. In fact, he starts with the residual representation $\bar\rho$, then associates some conductor $N$ to $\bar\rho$ (which may not be square free) and then proves that there is an isomorphism between the corresponding local ring of the Hecke algebra of tame level $N$ and a deformation ring attached to $\bar\rho$.
In Böckle's article, the restricted Hecke algebra is used instead of the full one. However, for the $R^{\mathrm{ord}}=T$ statement, this does not make a difference. For this, see Theorems 10 and 12 in the article by Gouvêa referenced below.