Dear William,
Hi. On page 5 of your notes
http://sage.math.washington.edu/simuw06/notes/notes.pdf
the correspondences between (a,b,c) and (x,y) are not as good as they
could be. If instead you use
f(a,b,c) = (n(a+c)/b, 2n^2(a+c)/b^2)
and
g(x,y) = ((x^2 - n^2)/y, 2xn/y, (x^2+n^2)/y),
which are inverses of each other, then there is sign-preservation: a,
b, and c are all positive if and only if x and y are both positive
(from y^2 = x^3 - n^2x = x(x^2 - n^2), if x and y are positive then so
is x^2 - n^2, so the first coordinate in g(x,y) is indeed positive).
Since this is a problem about lengths of sides of a triangle, it is
nice to have a correspondence which produces positive data from
positive data in both directions. In your correspondence, the (3,4,5)
right triangle is attached to the point (-3,9) on y^2 = x^3 - 36x. In
the above correspondence, the (3,4,5) right triangle is attached to
(12,36) instead.
The connection between the correspondences above and the ones in your
notes is this: the two sets each have an involution: (a,b,c) -->
(-a,-b,c) and (x,y) --> (-n^2/x, n^2y/x^2). If you apply f and g
above and then compose with these involutions, you get the
correspondences in your notes.
Best,
Keith
--
Oh, I was thinking you might change the notes! But perhaps you just
want to leave them as they really were. Rather than linking to my
email, consider instead a link to the wikipedia entry
http://en.wikipedia.org/wiki/Congruent_number, which I edited to
include this correspondence. It comes out of the intro to Tunnell's
paper (check it yourself). The only thing I really noticed is that
you can take out the absolute value signs in the correspondence as in
Tunnell's paper by just subtracting in the right order.