I don't know these topics very well, so please point out when I write things which are incorrect (I have a feeling this will happen). First I will give the definition I'm using of a primitive Hida family, second I will 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 $\mathcal{S}(N,\psi)$ be the space of 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 $\mathcal{S}(N,\psi)$. 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 $\varepsilon$ 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 $Np^{r}$. $F$ 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 of $\mathcal{C}^{ord}$ 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 $\mathbb{Q}_p$, $\mathcal{C}^{ord}(E)$ 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!