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Assume we have an Abelian varieties over the p-adic numbers, namely $ k=\mathbb{Q}_p$. Then the question is whether $A(k)$, the rational points over $k$, will form a p-adic analytic manifold.

I am reading a Book by Serre "Lie Algebras and Lie groups". i took the definition of Analytic manifolds from this text, I guess it should be an standard definition.

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Yes, this is true. For any smooth finite type $k$-scheme $X$, $k$ a non-Archimedean field, $X(k)$ has a canonical structure of $k$-analytic manifold in the sense of Serre's book (or Bourbaki). Usually nowadays one calls these locally $k$-analytic manifolds. I think this can be proved using "standard smooth affine opens" of $X$, or the Jacobian criterion for smoothness if you like, plus the inverse function theorem (proved in Serre's book).

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  • $\begingroup$ Thank you for the answer. I am lost a bit. In the Book Lie Algebras and Lie groups, Serre studies the properties of the topological space $X$ when X is Hausdorff, but $A(k)$ is not Hausdorff. btw, by Serre's book you mean the Lie algebra and Lie group one, am I right? $\endgroup$
    – Sina Baghal
    Commented Jan 21, 2015 at 17:52
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    $\begingroup$ The topology on $X(k)$ when regarding it as a locally $k$-analytic manifold is not the Zariski topology induced from $X$. I should have added "separated" to ensure that the topology on the $k$-points $X(k)$ (which is described in great detail in a paper of Brian Conrad which you can find on his website) is Hausdorff. And yes, I'm talking about Serre's book on Lie algebras and Lie groups. $\endgroup$
    – Keenan Kidwell
    Commented Jan 21, 2015 at 20:58

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