Math 346/646 - Homework 10 hints
1. Use Sage to compute the order of P . Then use a for loop to find a.
2. Here’s a series of steps I think you should follow.
(a) Take the first equation a2 + b2 = c2 + d2 and view it in affine space as a2 + b2 = c2 + 1. The
point (1, 0, 0) is on this surface. Parametrize the surface by drawing a line through (1, 0, 0) and
finding the second intersection point. (You can set a = xt + 1, b = yt and c = zt. This is a
parametrization of the generic line through (1, 0, 0).)
(b) Clear denominators in your parametrization to get a parametrization that works in projective space to a2 + b2 = c2 + d2 . You should have each of a, b, c and d equal to a homogeneous
degree two polynomial in x, y and z.
(c) Plug the parametrization from (b) into a3 + b3 − c3 − d3 . This gives you a homogeneous
degree 6 polynomial. Factor this polynomial using Sage. You should find some linear factors
and an irreducible degree 3 factor. Show that if any of the linear factors equals zero, then a = c
or a = d.
(d) The irreducible degree 3 factor defines a cubic curve C in P2 . Observe that P = (0 : 1 : 1)
is a point on this cubic. Use this to compute a map φ : C → E0 where E0 is some elliptic
curve. Use Sage to find a minimal model E1 of this curve.
(e) Find a rational point of order 2 on E1 and apply a translation to obtain a curve in the form
E2 : y 2 = x3 + ax2 + bx.
(f) Use the method of Chapter 3 to prove that the rank of E2 is zero.
(g) Use the Lutz-Nagell theorem to find all of the points of finite order on E2 , and (by (f)), all
of the rational points.
(h) Map these rational points from E2 → E1 → E0 → C and then back to a2 + b2 = c2 +
d2 , a3 + b3 = c3 + d3 . Make the desired conclusion.
3. (a) Note that (0, 0) is the only point of order two in E(Fp ).
(b) A for loop is not going to do it for you.
Q and hence
You want to use the idea from (a) repeatedly. You can use Sage to compute p+1
2
p+1
p+1
whether a is even or odd. If a is even, then a = 2a1 and then ( 4 )Q = ( 2 )a1 Q and using
this you can determine if a1 is even or odd. I hope that’s enough of a start.
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