Timeline for Notion of connected components for $\mathbb{Q}_p$-points of algebraic variety
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2023 at 18:29 | comment | added | Jacques | @LaurentMoret-Bailly I have in mind the way it works over $\mathbb{R}$. Even for an elliptic curve $E$, $E(\mathbb{R})$ can have several connected components with the usual topology. How can this be viewed from the point of view of algebraic variety? | |
| Oct 27, 2023 at 8:11 | comment | added | Laurent Berger | Have you taken a look at Brian Conrad's "Irreducible components of rigid spaces"? | |
| Oct 27, 2023 at 5:29 | comment | added | Laurent Moret-Bailly | Can you give an example of a connected $X$ such that (in your view) $X(\mathbb{Q}_p)$ should not be connected? | |
| Oct 26, 2023 at 13:49 | comment | added | Jacques | @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. | |
| Oct 26, 2023 at 11:51 | comment | added | YCor | 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. | |
| Oct 26, 2023 at 11:31 | comment | added | Jason Starr | With its metric topology, $\mathbb{Q}_p$ is totally disconnected. Are you asking about (rigid analytic) path connectedness? | |
| Oct 26, 2023 at 10:56 | history | asked | Jacques | CC BY-SA 4.0 |