I'm reading this article on strong approximation, and am coming across the following example of the hypersurface $X$ defined by $3x^3 + 4x^3 + 5z^3 = 0$.
The claim is that since any point on $X$ other than the origin is smooth, and since $X(\mathbb Z_p) \neq \emptyset$, $X(\mathbb Z_p)$ must be infinite.
My first question, is why this implies $X(\mathbb Z_p)$ is infinite. And second, what is meant by smooth points on $X$? Do we need to base change $X$ to $\mathbb Q$, or $\mathbb Q_p$, to talk about smoothness?
Any smooth $\Bbb Z_p$ variety is either empty or contains infinitely many points. This is a consequence of the implicit function theorem for $p$-adic analytic functions (see here for an online source, although I originally learned of it from a paper of Serre I can't find at the moment...). The implicit function theorem gives an p-adic analytic diffeomorphism (and in particular, a bijection) between an open subset of $X(\Bbb Z_p)$ and $\Bbb Z_p^n$, so the former has infinitely many points.
Secondly, smoothness is defined intrinsically for schemes and does not require base-changing. See for instance StacksProject on smoothness.