I'm trying to study Elliptic Curves currently, and in the book I'm reading there seems to be one question they keep gravitating around but never actually even attempt to answer, even though it seems like an obvious question within the context.
Does there exist a cubic protective curve that has no rational points but has points over $\mathbb{F}_p$ for every prime $p$?
I suspect this is an ill-formed question that stems from a misunderstanding of the subject, so if this question doesn't really make sense can you help me understand where it is wrong, and if not, is there something I'm missing here?