In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the eigencurve because I do not know any rigid analytic geometry. (I'm planning on learning some soon!) The following question is what I've come up with in my attempt to answer the question in the title. I cannot answer this question because I do not know enough about the eigencurve.

In what follows please point out when I write things which are incorrect, for I have a feeling that not everything I write is 100% correct, but I think the question is well posed.

I will construct a primitive Hida family, say what I know about the ordinary locus of the eigencurve, and then I will ask a question about the relationship between the two.

Let $\mathcal{S}^o(N,\psi)$ be the space of ordinary cuspidal Hida families of tame level $N$ and character $\psi$. Let $R$ be the ring obtained by adjoining the values of $\psi$ to $\mathbb{Z}_p$. Let $\mathbb{T}_{N,\psi}^{new}$ be the new quotient of the Hecke algebra acting on $\mathcal{S}^o(N,\psi)$. By a theorem of Hida, $\mathbb{T}_{N,\psi}^{new}$ is a finite, torsion free, $\Lambda_R = R[[1 + p\mathbb{Z}_p]]$-algebra. Then, letting $\mathscr{L}_R$ be the fraction field of $\Lambda$, $\mathbb{T}_{N,\psi}^{new}\otimes_{\Lambda_R}\mathscr{L}_R$ is a finite product of finite field extensions of $\mathscr{L}_R$. Let $\mathbb{I}_{\mathscr{L}}$ be one of the fields showing up in the product and let $\mathbb{I}$ be the integral closure of $\Lambda_R$ in $\mathbb{I}_\mathscr{L}$. The image of the map $$\mathbb{T}_{N,\psi}^{new}\longrightarrow \mathbb{T}_{N,\psi}^{new}\otimes_{\Lambda_R}\mathscr{L}_R\longrightarrow\mathbb{I}_{\mathscr{L}}$$ lands in $\mathbb{I}$. Let $a_n\in\mathbb{I}$ be the image of the $n$th Hecke operator. Then $$F : = \sum_{n\geq 1} a_nq^n\in\mathbb{I}[[q]]$$ has the property that for all continuous $R$-algebra homomorphisms $\nu : \mathbb{I}\longrightarrow \overline{\mathbb{Q}}_p$ such that $\nu([1+p]) = \varepsilon(1+p)(1+p)^k$ where $[1+p]\in\Lambda_R\subset\mathbb{I}$ is the group like element associated to $1+p\in 1 + p\mathbb{Z}_p$, $\varepsilon$ is a finite character of $\mathbb{Z}_p^\times$ of conductor $p^r$, and $k\geq 2$ is an integer, $$\nu(F): = \sum_{n\geq 1}\nu(a_n)q^n$$ is a Hecke eigenform of weight $k$, level $Np^{r'}$ where $r' = \max(r,1)$, and character $\psi\varepsilon\omega^{-k}$ where $\omega$ is the Teichmuller character, which is new at level $Np^{r}$ and has $U_p$ eigenvalue a $p$-adic unit. $F$ is the primitive Hida family that my question is about.

Let $\mathcal{C}^{ord}$ be the ordinary locus of the eigencurve of tame level $N$ and character $\psi$. By definition $\mathcal{C}^{ord}$ is a rigid analytic $\mathbb{Q}_p$-variety, and I don't know what this means. None the less, it is my understanding, that there is a rigid analytic $\mathbb{Q}_p$-subvariety $\mathcal{C}_F\subset\mathcal{C}^{ord}$ corresponding to the Hida family $F$, which is a connected component of $\mathcal{C}^{ord}$. By being a rigid analytic $\mathbb{Q}_p$-variety, for any field extension, $E$, of $\mathbb{Q}_p$, the $E$-points of $\mathcal{C}^{ord}$ and $\mathcal{C}_F$, $\mathcal{C}^{ord}(E)$ and $\mathcal{C}_{F}(E)$, are topological spaces with $p$-adic topologies.

My question is about the relationship between the sets $\mathcal{C}_{F}(\overline{\mathbb{Q}}_p)$ and $Hom_{cont,R-alg}(\mathbb{I},\overline{\mathbb{Q}}_p)$. Specifically, is the map $$\begin{array}{rcl} Hom_{cont,R-alg}(\mathbb{I},\overline{\mathbb{Q}}_p) &\longrightarrow &\mathcal{C}_F(\overline{\mathbb{Q}}_p)\\ \nu &\longmapsto &\nu(F)\end{array}$$ well-defined? If it is well defined, is it a bijection? Finally, if it is a bijection, is there a topology that we can put on $Hom_{cont,R-alg}(\mathbb{I},\overline{\mathbb{Q}}_p)$ without making reference to the eigencurve, such that the above map is a homeomorphism of topological spaces?

Any help with any of these questions, help that would further my understanding of what I'm talking about, or references which explain the precise relationship between Hida families and the eigencurve would be greatly appreciated!

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    $\begingroup$ Just a comment: firstly, there is a construction called "rigid generic fibre" which produces rigid spaces from p-adic formal schemes; $\mathcal{C}^{\mathrm{ord}}$ is the generic fibre of the formal scheme $\mathrm{Spf} \mathbf{T}_{N, \psi}$, where $\mathbf{T}_{N, \psi}$ is the Hecke algebra acting on $\mathcal{S}^{o}(N, \psi)$ in your notation. So your question actually has nothing to do with rigid geometry as such -- you can reduce it entirely to questions about formal schemes. $\endgroup$ – David Loeffler May 31 '16 at 7:23
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    $\begingroup$ Secondly, the p-adic topology is generally a fairly terrible topology to put on the points of a rigid space or formal scheme, just as in the case of an algebraic variety. The Zariski topology is a much better one to use here. $\endgroup$ – David Loeffler May 31 '16 at 7:25
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    $\begingroup$ With that in hand, the answer to your question is this. The space $\mathcal{C}_F$ is an irreducible component of $\mathcal{C}^{\mathrm{ord}}$, and the field of meromorphic functions on it is $\mathbf{I}_{\mathscr{L}}$ [seriously, how many LaTeX alphabets can one use in one MO question?]. So $Spec \mathbf{I}$ is the normalisation of $\mathcal{C}_F$, and in particular there is a canonical map $Spec \mathbf{I} \to \mathcal{C}_F$ which is an isomorphism away from finitely many points. $\endgroup$ – David Loeffler May 31 '16 at 7:29
  • $\begingroup$ David, thanks a lot for this! Sorry for all the latex alphabets... I hear what you are saying, but let me ask a follow up question. $\endgroup$ – Will Dukeminier May 31 '16 at 22:40
  • $\begingroup$ My interest in the topology stems from trying to understand what it means for a two-variable $p$-adic $L$-function to have one of its variables an open subset of the eigencurve. The case I'm interested in is when the open subset contains a point corresponding to a weight one modular form, which is smooth, but not necessarily `{e}tale over weight space. Is it possible to say what it means for a function to be analytic on the open subset in question, by making reference only to the set $Hom_{cont,R-alg}(\mathbb{I},\overline{\mathbb{Q}}_p^\times)$ and some $p$-adic structure on this set? $\endgroup$ – Will Dukeminier May 31 '16 at 22:42

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