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. 



Abelian varieties as analytic manifolds
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). 


