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!

### Like this:

Like Loading...

*Related*

I have a vague souvenir of boundedness of ranks being mentioned as suggested/conjectured by Shioda or Honda. This would have been something I heard around 1996 during my PhD; I don’t remember any reference, however, but I’ll try to dig in my unreliable memory…

That was before 1967 when Tate and Šafarevič proved the rank is unbounded over k(t) for each finite field k.

By 1966, Cassels could state “it has been widely conjectured that there is an upper bound for the rank depending only on the groundfield” though he gives no citation and then says “this seems to me implausible” (meaning conjecturing from scanty data, or the rank boundedness itself?) noting that high rank could only occur with large coefficients in any event. At the end of paragraph he states: “We shall give another reason for supposing that the rank is unbounded at the end of S26”, where the finiteness of ш is discussed (I guess he implies that you can get either large rank or large ш, and figures one is no more likely than the other?). Definitely historically, it seems that the 60s were the time when the tipping went to unbounded rank.

See pg257 (pg65 of the PDF) of the survey article “Diophantine Equations with Special Reference to Elliptic Curves” http://jlms.oxfordjournals.org/content/s1-41/1/193.full.pdf

I got this quote from a MathOverflow comment of Milne.