Example of a non-smooth irreducible component of the generic fibre of a Hida family?

Question

Is there a known example of a non-smooth irreducible component of the rigid generic fibre of a Hida family?

Let me explain some of the context around this question (but I'm not going to explain Hida theory -- he wrote a book on it if anyone is interested!). Let $\Lambda$ be the ring $\mathbf{Z}_p[[T]]$ as is usual in Hida theory. Given a level $N$ and a prime $p$ Hida constructs a semi-local $\Lambda$-algebra $\mathbf{T}$ interpolating ordinary Hecke algebras of level $N$ and all weights (in a congruence class mod $p-1$ (or mod $2$ if $p=2$)), and this algebra is typically finite and free as a $\Lambda$-module. In particular it gives rise to a finite flat cover of the rigid analytic generic fibre of $\Lambda$, which is a $p$-adic open disc. Let me call this rigid space the "Hida generic fibre".

One might ask what Hida generic fibres can look like. Certainly they have finitely many irreducible components. Examples are known where these components map down in a ramified way down to weight space; examples are known where two irreducible components cross, giving rise to singularities in the generic fibre. Components crossing can happen for several reasons, for example a component which is generically new at some auxiliary prime $\ell$ dividing $N$ once can meet a component which is generically old at $\ell$, a CM family can meet a non-CM family, or two CM families corresponding to different imaginary uadratic fields can meet (in a weight 1 point for example). All of these give examples of phenomena where the map from the Hida rigid space down to weight space is not etale (and although my memory is getting rusty these phenomena might be all the examples I know of how the map down to weight space can fail to be etale). But in all these cases it could happen that every component of the Hida generic fibre is smooth. I don't know any examples where a component is not smooth. Does anyone?

Answer 1

This is an answer of rather low quality, but let me report that in several discussions about this topic with (many) experts over the span of many years, I have neither met anyone knowing of such an example. Apart from computing the irreducible components of the (abundant) non-Gorenstein Hecke algebras $\mathbf T_{\mathfrak m}$ when $p=2$, no one I met seemed to have a clear idea how to look for an example either (or for a proof that they do not exist). When $p\neq 2$, Hida himself firmly believes the irreducible components are always smooth and once explained to me a believable heuristic argument based on the (conjectured) semi-simplicity of Frobenius action. As for other natural questions about the properties of irreducible components of Hida families, this might be hard.