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!
