Let $ k$ be a local field of characteristc zero. I’m interesting in understanding how morphisms of schemes of finite type over $ k$ become morphisms of analytic manifolds on passing to rational points. For example, if $ Z$ is a smooth closed subscheme of $ \mathbb A^n_k$ , then $ Z(k) \rightarrow k^n$ becomes a closed (embedded) submanifold. This directly follows from the constant rank theorem from differential geometry and the Jacobian criterion for smoothness.
Now, let $ G$ be a geometrically connected affine algebraic group over $ k$ and $ X$ an irreducible affine variety over $ k$ on which $ G$ acts. Suppose that the geometric quotient $ X/\sim$ of $ X$ under the action of $ G$ exists. I’m using these notes for the definition of geometric quotient. Can we say that map on rational points
$ $ X(k) \rightarrow X/\sim(k)$ $
is a submersion of manifolds, that $ X/\sim(k) = X(k)/\sim$ , and that $ X/\sim(k)$ and is the quotient of $ X(k)$ by $ G(k)$ in the category of analytic manifolds over $ k$ ? I do not know of any references which discuss this sort of thing.
My reference for analytic manifolds over local fields is Serre, Lie Groups and Lie Algebras. The following result from Serre might be useful: “If $ X$ is a manifold, and $ \sim$ an equivalence relation on $ X$ , there is at most one manifold structure on the quotient space $ X/\sim$ making the quotient map a submersion of manifolds. Then $ X \rightarrow X/\sim$ is a quotient map in the category of manifolds.
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