# On the structure theory of the group of rational points of an algebraic group defined over a local non-Archimedean field

Let $$k$$ be a non-Archimedean local field, and let $$G$$ be an affine algebraic group over $$k$$. The literature suggests that the following are well established facts:

The group $$G(k)$$ of $$k$$-rational points in $$G$$ inherits the structure of a locally profinite group w.r.t. the subspace topology, and in the case that $$char(k)=0$$ the group $$G(k)$$ also has the structure of an analytic manifold, turning it into a $$p$$-adic Lie group.

Unfortunately, I am not well versed in the theory of analytic manifolds, and its apparent interplay with algebraic geometry in the above setting. Taking this into account, I would like to ask clarifications and/or references for the above mentioned facts (which to me are folklore for now) and the following questions:

Keep the notation introduced so far. In what follows, $$\overline{k}$$ denotes the algebraic closure of $$k$$. Furthermore assume that $$char(k)=0$$.

1) What is the relation between the subspace topology on $$G(k)$$ inherited from $$G(\overline{k})$$ and the topology of its underlying analytic manifold?

2) More generally, if $$V$$ is an affine variety over $$k$$, does $$V(k)$$ possess the structure of an analytic manifold in any canonical way? If so, what is the relation between its underlying topology and the subspace topology inherited from $$V(\overline{k})$$?

3) Suppose $$G$$ acts $$k$$-morphically on an affine variety $$V$$ over $$k$$. The action morphism $$G \times V \rightarrow V$$ gives rise to an action map $$G(k) \times V(k) \rightarrow V(k)$$. Is this map continuous if both $$G(k)$$ and $$V(k)$$ are given the subspace topology inherited from $$G(\overline{k})$$ and $$V(\overline{k})$$ respectively, and $$G(k) \times V(k)$$ is given the product topology? In case question "2)" has a positive answer, what can be said about the same action map if $$G(k)$$ and $$V(k)$$ are both given the analytic manifold topology and $$G(k) \times V(k)$$ is given the product topology?

Thank you in advance.

• (1) it's hard to answer if you don't say which topology you consider on $G(\bar{k})$. (2) yes if $V$ is smooth for the 1st question. (And yes to the 2nd if you endow $\hat{k}$ with the norm extending a norm on $k$ and topologize $V(\hat{k})$ accordingly) (3) again it's hard to answer but if varieties are endowed with the norm topology then regular maps are continuous, and this applies to the map $G(k)\times V(k)\to V(k)$; if $V$ is smooth this is also a $k$-analytic map. – YCor Nov 1 '19 at 0:56
• I think that most of your questions are answered in Bjorn Poonen's book Rational points on varieties. I would look there. – Alex Youcis Nov 1 '19 at 9:49