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MATH 4450/
HOMEWORK 7
DUE 3/19/
Do the following book problems (for 11(d), you can write down the proof I gave in class in your own words, if you don’t want to use Lagrange’s theorem):
16.7: 3, 5, 11, 12, 15, 16, 17 (6450 students only), 19, 20, 21.
16.8: 3, 4.
Your instructions for the extra problems this week are: 4450 students should do Extra Problems 1, 2, and 5. As usual, 6450 students have to do everything.
Extra Problem 1: Let p be an odd prime, and a ∈ F∗ p. Recall that an elliptic curve over Fp is called supersingular if the number of Fp-points on it equals p + 1. (Actually, this is not quite right when p = 2 or 3, because 1 and 2p + 1 fall inside the Hasse bound there as well; but at least for p ≥ 5 this definition is correct). Let E be the elliptic curve over Fp given by the equation y^2 = x^3 + ax.
(a) Suppose that p ≡ 3 mod 4. Show that for any x ∈ F∗ p, x^2 6 ≡ −a mod p, exactly one element of the set {x, −x} is the x-coordinate of a point on E.
(b) Conclude that if p ≡ 3 mod 4, E is supersingular. (You’ll probably want to consider the cases where −a is or is not a square mod p separately. Don’t forget ∞ when you count points!)
(c) Now suppose that p ≡ 1 mod 4. Let α be an element of Fp such that α^2 ≡ −1 mod p. (Why does this α exist?) Show that if (x, y) ∈ E(Fp), then (−x, αy) ∈ E(Fp) as well.
(d) Conclude that if p ≡ 1 mod 4 and
a p
= 1, then #E(Fp) is divisible by 4, so E can’t
be supersingular. (Hint: most of the points come in sets of four, because of part (c); then there are some special points where y = 0 that you haven’t counted yet, and don’t forget ∞!)
Remark: It is true that E can’t be supersingular in part (d) even if
a p
= −1, but I don’t
know an elementary proof.
Extra Problem 2: Let E be the elliptic curve y^2 = x^3 + 7. We will show that there are no points (x, y) on E where x and y are both integers. So from now on, assume that y^2 = x^3 + 7 with x and y both integers.
(a) Show that x is odd. (b) Show that there is a prime p ≡ 3 mod 4 that divides x^2 − 2 x + 4.
(c) Show that p|y^2 + 1. This gives a contradiction. (Why?) Remark: In fact there are no rational solutions to the equation y^2 = x^3 + 7.
Extra Problem 3: Let K be a field (of characteristic not equal to 2 or 3), and a ∈ K∗. Consider the curve C over K defined by the equation X^3 + Y 3 = aZ^3 in two-dimensional projective space. Let O = (1 : − 1 : 0) ∈ C. Define an addition law on this curve by the familiar-looking rule that three points are deemed to sum to O if and only if they are collinear (and O is taken to be the identity element with respect to this addition).
(a) To warm up, show that if P = (b : c : d) ∈ C, then the negative of P with respect to the above addition law is the point (c : b : d) ∈ C.
(b) Let E be the elliptic curve y^2 = x^3 − 432 a^2. Consider the functions f and g defined by the following rules:
f (X : Y : Z) =
12 a
Z
X + Y
, 36 a
X − Y
X + Y
g(x, y) = (36a + y : 36a − y : 6x) Show that f is a function from C(K) to E(K) (if we make the reasonable-looking con- vention that f (O) = ∞). Show that g is a function from E(K) to C(K) (if we make the reasonable-looking convention that g(∞) = O). Show that f and g are inverses. (Remember that in projective space, scaling all the coordinates by the same constant doesn’t change the point.)
(c) Consider the elliptic curve E 1 given by the equation y^2 = x^3 − 432 over Q. Determine the set E 1 (Q). (You might want to assume some sort of ”last theorem” or something...)
(d) Consider the curve C 9 in P^2 given by the equation X^3 + Y 3 = 9Z^3. Certainly the point P = (1 : 2 : 1) is on this curve. Find a rational point on C 9 which is not (1 : − 1 : 0), (1 : 2 : 1), or (2 : 1 : 1). (Hint: Consider g(2f (P )).)
Remarks: If you believe that f and g preserve collinearity (which is not hard to see from the formulas), then you can see that the addition law we’ve defined on C has all the properties of the addition law on E (it inherits them via the bijection g). Using this, it is not terribly difficult to write down the formulas for the sum of two points on C (or you can do it directly from the definitions, of course).
Extra Problem 4: Let C be the curve in P^3 given by the equations u^2 + v^2 = w^2 u^2 − v^2 = z^2 Let E be the elliptic curve y^2 = x^3 − 16 x. (a) Show that the transformations
f (u : v : w : z) =
4(w − z) 2 u − w − z
16 v 2 u − w − z
g(x, y) = (16 + x^2 : 4y : x^2 + 8x − 16 : x^2 − 8 x − 16)
