## How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?

I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties.

In the literature (e.g. in Coleman-Mazur) I've seen people talk about the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$ (here $\mathbf{C}_p$ is the completion of an algebraic closure of $\mathbf{Q}_p$). Based on the case of schemes, if $X$ and $T$ are rigid spaces over $\mathbf{Q}_p$, I assume the set of $T$-valued points of $X$ is the set $\mathrm{Hom}_{\mathbf{Q}_p}(T,X)$, where the morphisms are in the category of $G$-topologized spaces over $\mathrm{Sp}(\mathbf{Q}_p)$. But $\mathbf{C}_p$ is not a $\mathbf{Q}_p$-affinoid algebra (at least I don't think it is), so, strictly speaking, it's associated $G$-topologized space $\mathrm{Sp}(\mathbf{C}_p)$ is not a rigid space over $\mathbf{Q}_p$, right? I guess there is a morphism of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}_p)\rightarrow\mathrm{Sp}(\mathbf{Q}_p)$, and so one could define the $\mathbf{C}_p$-points of $X$ in the same way as I did above, just morphisms of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}_p)\rightarrow X$ compatible with the morphisms to $\mathrm{Sp}(\mathbf{Q}_p)$.

Question: Is this what is meant by the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?

The only alternative I can think of is to define the $\mathbf{C}_p$-points of a rigid space $X$ over $\mathbf{Q}_p$ as the $\mathbf{C}_p$-valued points of the base change $X_{\mathbf{C}_p}$. Although, if my intuition drawn from schemes is to be trusted, this is probably equivalent to the definition I suggested above.

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Yes, and a "morphism" $X' \rightarrow X$ from a rigid-analytic space over $k'$ to one over $k$ for an extension of non-archimedean fields $k'/k$ means a morphism $X' \rightarrow X_{k'}$. This viewpoint is very convenient with Berkovich spaces (where there is change of ground field much more often than in rigid-analytic geometry). Sometimes you'll see $\mathbf{P}^1(\mathbf{C}_p)-\mathbf{P}^1(\mathbf{Q}_p)$ as a synonym for $\mathbf{P}^{1,{\rm{an}}}_{\mathbf{Q}_p} - \mathbf{P}^1(\mathbf{Q}_p)$ in settings where the $\mathbf{Q}_p$-structure matters, and that is simply wrong. – xuhan Nov 25 at 19:50
@Xuhan: Not at all: 1) On $\mathbf P^{1,\mathrm{an}}_{\mathbf Q_p}$, points of $\mathbf P^1(\mathbf C_p)$ which are conjugate over $\mathbf Q_p$ are identified; 2) And the Berkovich space has many more points than these ones. – ACL Nov 25 at 20:29
Dear ACL: I agree with your (1) but disagree with your "Not at all" because transcendental points of $\mathbf{P}^1(\mathbf{C}_p)$ relative to $\mathbf{Q}_p$ do not "correspond" to any points in $\mathbf{P}^{1,{\rm{an}}}_{\mathbf{Q}_p}$; it makes the abuse of notation seem akin to considering a finite type $\mathbf{Q}$-scheme $X$ and writing $X(\mathbf{C})$ when we mean $X$. Also, my intent when I mentioned Berkovich spaces is that the idea of a map from a $k'$-analytic object to a $k$-analytic one tends to arise more often there than in rigid-analytic geometry precisely because of your (2). – xuhan Nov 25 at 21:09
Dear ACL: To clarify further, by "$\mathbf{Q}_p$-structure" I had in mind coherent sheaves and morphisms, not just points. I'm not aware of a useful descent theory for rigid-analytic geometry relative to ground field extensions like $\mathbf{Q}_p \rightarrow \mathbf{C}_p$, so to make constructions respecting $\mathbf{Q}_p$-structure I believe one has to work with geometric objects over $\mathbf{Q}_p$ (or at least finite extensions of it, and then use Galois descent). It is this sense in which writing $X(\mathbf{C}_p)$ as proxy for $X$ seems to be a problematic abuse of notation. – xuhan Nov 26 at 1:45

I think in Coleman-Mazur, it is just to be taken in the sense of $\mathbb C_p$-points of $X_{\mathbb C_p}$. This is compatible with the more abstract definition you propose. Indeed, as $\mathrm{Sp}(\mathbb C_p)$ is just one point, it is enough to work with an affinoide space $X$ associated to an affinoide algebra $A$ over $\mathbb Q_p$. Then it is clear that the continuous homomorphisms from $A$ to $\mathbb C_p$ coincide with continuous homomorphisms from $A\widehat{\otimes}_{\mathbb Q_p} \mathbb C_p$ to $\mathbb C_p$.