Math 346/646 - Homework 9 hints or P

Math 346/646 - Homework 9 hints
1. Consider the case that P1 or P2 is equal to T separately. Otherwise, it may be helpful to
use the formula (on page 80 in edition 1) for (x, y) + (0, 0).
2. (a) This should be similar to the example in class where the rank is zero.
(b) The rank is positive.
(c) This curve has a lot of points.
3. Just use the method we have for computing |im α| and |im α|. It may be helpful to remember
that the points (p, 0), (0, 0), (−p, 0) are on C. It will turn out that im α = {±1, ±p}.
On the other hand im α will be {1}. Showing the others aren’t possible basically boils down
to the equations
N 2 = 2M 4 + 2p2 e4 and N 2 = 4M 4 + p2 e4 and N 2 = 2pM 4 + 2pe4 .
In the first case, look at the equation modulo 16 and the in the second and third cases, you
can get a contradiction using the fact that x2 ≡ −1 (mod p) is not solvable.