According to Silverman:
In order to show that an algebraic set $V/\mathbb{Q}$ has no $\mathbb{Q}$-rational points, it suffices to show that the corresponding homogeneous polynomial equations have no nonzero solutions modulo $p$ for any one prime $p$ (or even for one prime power $p^r$). A more succinct way to phrase this is to say that if $V(\mathbb{Q})$ is nonempty, then $V(\mathbb{Q}_p)$ is nonempty for every $p$-adic field $\mathbb{Q}_p$.
(The Arithmetic of Elliptic Curves, p.8.)
I don't see how the two statements are related:
Question. What's going on here?