4
$\begingroup$

Is there an interesting notion of connected components for the $\mathbb{Q}_p$-points of an algebraic variety over $\mathbb{Q}_p$? By "interesting" I mean a notion satisfying the following. Given a connected algebraic variety $X$ over $\mathbb{Q}_p$, I would like $X(\mathbb{Q}_p)$ to have finitely many connected components but not always one, similarly to what happend over $\mathbb{R}$.

An idea would be to embed $X(\mathbb{Q}_p)$ into the analytification of $X$ and take in some sense an induced topology as we do over $\mathbb{R}$. For instance one can define $X(\mathbb{Q}_p)$ to be disconnected if there is an admissible cover ${U,V}$ in $X^{an}$ whose union contains $X(\mathbb{Q}_p)$, whose intersection contains no point of $X(\mathbb{Q}_p)$ and such that each $U$ and $V$ intersect $X(\mathbb{Q}_p)$ non-trivially. Could this work?

Improve this question
$\endgroup$
6
  • $\begingroup$ With its metric topology, $\mathbb{Q}_p$ is totally disconnected. Are you asking about (rigid analytic) path connectedness? $\endgroup$
    – Jason Starr
    Commented Oct 26, 2023 at 11:31
  • $\begingroup$ A motivating example might be the following. When $G$ is a simple isotropic algebraic group, the subgroup of $G(\mathbf{Q}_p)$ generated by unipotents has finite index and is a natural candidate, to say that its cosets are the connected components. However, when $G$ is anisotropic, this subgroup is trivial, the group $G(\mathbf{Q}_p)$ is profinite and I have no idea what would be a choice of "unit connected component", except maybe the whole group. $\endgroup$
    – YCor
    Commented Oct 26, 2023 at 11:51
  • 1
    $\begingroup$ @JasonStarr I am looking for a notion of connectedness, not path connectedness. Even though $\mathbb{Q}_p$ is totally disconnected, rigid analytic theory have a nice notion of connectedness. As I understand it, the analytification of a connected algebraic variety will be connected. I am looking for a notion that could lead different components even for connected algebraic variety. This happends when you look at $\mathbb{R}$-points of an algebraic variety with the Euclidean topology for instance. $\endgroup$
    – Jacques
    Commented Oct 26, 2023 at 13:49
  • 2
    $\begingroup$ Can you give an example of a connected $X$ such that (in your view) $X(\mathbb{Q}_p)$ should not be connected? $\endgroup$
    – Laurent Moret-Bailly
    Commented Oct 27, 2023 at 5:29
  • $\begingroup$ Have you taken a look at Brian Conrad's "Irreducible components of rigid spaces"? $\endgroup$
    – Laurent Berger
    Commented Oct 27, 2023 at 8:11

0

Reset to default

You must log in to answer this question.