Are there elliptic curves with small rank?

Usually when people muse about distribution of ranks of elliptic curves, they’re wondering how large the rank of an elliptic curve can be. But as Tim Dokchitser pointed out to me, there are questions in the opposite direction which are equally natural, and equally mysterious. For instance: we do not know that, for every number field K, there exists an elliptic curve E/K whose Mordell-Weil rank is less than 100. Isn’t that strange?

Along similar lines, Bjorn Poonen asks: is it the case that, for every number field K, there exists an elliptic curve E/Q such that E(Q) and E(K) have the same positive rank? Again, we don’t have a clue. See Bjorn’s expository article “Undecidability in Number Theory” (.pdf link) to find out what this has to do with Hilbert’s 10th problem.