From: brock@ccr-p.ida.org (Bradley Brock) Newsgroups: sci.math Subject: Re: Does FLT hold for the Gaussian integ Date: 13 Jul 1995 23:53:07 -0400 Keywords: elliptic curves In article <3sk5i5\$765@lyra.csx.cam.ac.uk>, cet1@cus.cam.ac.uk (Chris Thompson) writes: |> In article <9506181512591.kevin2003.DLITE@delphi.com> kevin2003@delphi.com (Kevin Brown) writes: |> As long as we are talking about the case n=3, this is a question about |> elliptic curves. The curve x^3+y^3=z^3 (a.k.a. y^2 = x^3-432, or as |> y^2+y = x^3-7) has no points defined over Q except the torsion points |> (0,1,1), (1,0,1), and the point at infinity (1,-1,0). The question |> being raised here is, essentially, "for which quadratic extensions |> of Q does this curve have positive rank?" |> |> Not that this enables me to answer it... where is Noam Elkies when we need |> him? This is equivalent to asking which quadratic twists of this curve have positive rank. Last year Elkies actually wrote a paper on a related subject in the ANTS proceedings (ftpable from Harvard). He was trying to answer when does the cubic twist of this curve x^3+y^3=az^3 (a.k.a. y^2=x^3-432a^2) have positive rank. The analytic rank has odd parity when a=4,7,8 mod 9 and a is prime, so in these cases conjecturally the cubic twist has positive rank. This was proved for a=4,7 mod 9. Furthermore, I believe Zagier has done computations that indicate the curve has rank 2 roughly 1/8 of the time. -- Bradley Brock, IDA/CCR-P, Thanet Road, Princeton, NJ 08540 brock@ccr-p.ida.org,brock@alumni.caltech.edu,609-924-3061(fax) "Football exemplifies the worst features of American life: it's violence punctuated by committee meetings."--George Will