From: rusin@math.niu.edu (Dave Rusin)
Subject: Re: Still another elliptic curve
Date: 30 Dec 1998 07:16:27 GMT
Newsgroups: sci.math.research
Keywords: Yet another elliptic surfaces

Hauke Reddmann <fc3a501@AMRISC02.math.uni-hamburg.de> wrote:
>y**2=s**4*(u**2-1)+4*s**3*u*(-3*u**2+4)+2*s**2*
>     (2*u**4-23*u**2-1)+4*s*u*(3*u**2-4)+u**2-1
>
>It's easy to take u as the second variable and
>reduce to normal form (conveniently the u**4
>term is already a square), but I want s as the
>second variable. Anyone out to find a rational point
>y=f(u),s=g(u)?

I'm not sure what you expect  f  and  g  to be (rational functions?)
but at the very least you seem to expect there to be point (y,s)
in each section  u=u0. This is not the case, and indeed the values
of  u0  for which such a point exist seem to be rare. I tried the
first few integral values of  u0. In most cases there are no
rational points on this curve because there aren't even any p-adic
points for one or more of p=2, 3, 5, or 7. For u0=13,17,25,29 there
were no p-adic restrictions (the curve is everywhere locally solvable)
but a very brief search turned up no points. Only for u0=1 was there
(obviously!) a point, but with this value for  u  we get an elliptic
curve whose rank over the rationals is  0;  there are only two
rational points (one at infinity).

dave
==============================================================================

From: Peter-Lawrence.Montgomery@cwi.nl (Peter L. Montgomery)
Subject: Re: Still another elliptic curve
Date: Wed, 30 Dec 1998 07:46:06 GMT
Newsgroups: sci.math.research

In article <76autt$1sn$1@rzsun02.rrz.uni-hamburg.de> 
fc3a501@AMRISC02.math.uni-hamburg.de (Hauke Reddmann) writes:
>y**2=s**4*(u**2-1)+4*s**3*u*(-3*u**2+4)+2*s**2*
>     (2*u**4-23*u**2-1)+4*s*u*(3*u**2-4)+u**2-1

>It's easy to take u as the second variable and
>reduce to normal form (conveniently the u**4
>term is already a square), but I want s as the
>second variable. Anyone out to find a rational point
>y=f(u),s=g(u)?



# Maple program follows
ysq := s**4*(u**2-1)+4*s**3*u*(-3*u**2+4)+2*s**2*
     (2*u**4-23*u**2-1)+4*s*u*(3*u**2-4)+u**2-1;    # Reddmann's right side
a := s - 1/s - 6*u + 2*u/(u^2 - 1);
factor(ysq/s^2 - (u^2 - 1)*a^2);      # Depends only on u

#    This factors as -4*(2*u^2 - 1)^3 / (u^2 - 1).
#    Try u = 2.  We must write -1372/3 as square - 3*square.
#    This has no rational solutions since 3 is a quadratic non-residue mod 7.

#    A solution over Q(u) might have u-2 in the denominators.
#    But one can find other values of u with this problem.

;quit;
-- 
        Peter-Lawrence.Montgomery@cwi.nl    San Rafael, California
The bridge to the 21st century is being designed for the private autombile.
The bridge to the 22nd century will forbid private automobiles.