Until I get a chance to write this out in more detail, the only summary
I will place here of the higher-descent methods will be the letter from
Siksek first explaining this to me. (Gee, it's been a busy 6 weeks!)


From S.Siksek@ukc.ac.uk Thu Jan 23 07:35:17 1997

Dear Prof Rusin,
I am afraid I have no software to do the 4-descents. However, for the
homogeneous spaces arising from descent via 2-isogeny you don't need any
number field arithmetic to do further descents.
Lets say your homogenous space is
 y^2= a*x^4+b*x^2*z^2+c*z^4
where a,b,c are integers (this is the shape of the hgs spaces arising
from descents via 2-isogeny). Lets suppose its everywhere locally soluble.
Then the corresponding curve of genus 0
Y^2=a*X^2+b*X^2*Z^2+c*Z^2
is everywhere locally soluble. Thus it has a solution in integers and we
can parametrize the solutions (if you write this in diagonal form you
can parametrize the solutions (if any) using the maple command isolve).
We will get that
X:Z:Y=q_1:q_2:q_3
where q_1,q_2,q_3 are hgs quadratic forms in say u,v. We can let
X=x^2, Z=z^2; where we can assume that x,z are coprime. Further we may assume
that u,v are coprime. Then we know that
d*x^2=q_1(u,v)
d*z^2=q_2(u,v)
where d is an integer dividing q_1(u,v) and q_2(u,v). We denote this curve by
C_d. Since u,v are coprime we know that d must divide the resultant of q_1 and
q_2. Hence we have finitely C_d to worry about. We can eliminate all those that
are not locally soluble. Again we can parametrize the solutions of the first
and substitute into the second. Say if we have the u:v:x=p_1:p_2:p_3 are
the parameteric solutions of the first equation where the p's are quadratic
forms in r,s say. Then after scaling and substituting in the second equation we
get
d*z^2=q_2(p_1,p_2)=f(r,s)
where f is a binary quartic form in r,s. Thus we have carried out a descent
without going through any numberfield stuff.

I hope the above is not too obscure; it might be helpful for you to look at
the papers by Bremner and Cassels, and by Bremner on y^2=x(x^2+p) which are
referred to in the 4-descents paper. Please feel free to ask more questions if
it still doesn't make sense.