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:
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