Bounded rank was probable in 1950

Somehow I wrote that last post about bounded ranks without knowing about this paper by Mark Watkins and many other authors, which studies in great detail the variation in ranks in quadratic twists of the congruent number curve.  I’ll no doubt have more to say about this later, but I just wanted to remark on a footnote; they say they learned from Fernando Rodriguez-Villegas that Neron wrote in 1950:

On ignore s’il existe pour toutes les cubiques rationnelles, appartenant a un corps donné une borne absolute du rang. L’existence de cette borne est cependant considérée comme probable.

So when I said the conventional wisdom is shifting from “unbounded rank” towards “bounded rank,” I didn’t tell the whole story — maybe the conventional wisdom started at “bounded rank” and is now shifting back!

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.