Page 283 -

260 13. Elliptic Curves over Finite Fields

(3.1) Definition. An elliptic curve E defined by a cubic equation f (w, x, y) over a
field k of characteristic p is supersingular provided the coefficient of (wxy) p−1 in
f (w, x, y) p−1 is zero.

A supersingular elliptic curve is also said to have Hasse invariant 0 or height 2,
otherwise the curve has nonzero Hasse invariant or height 1. The origin of these other
terminologies will be clearer later. The concept was first studied by Hasse [1934] and
it was referred to as the invariant A.

(3.2) Example. For p = 2 the equation of any elliptic curve E can be put in the
following form relative to a point of order 3:

2 2 3
E α : wy + w y + αwxy = x .
Since p − 1 = 2 − 1 = 1, it follows that E α is supersingular for exactly one curve
when α = 0, that is, the curve

2 3
y + y = x .
This is the curve considered in (1.3) where the number of points is p + 1 = 2 + 1.
This property is characteristic of supersingular curves defined over the prime field as
we will see in §4.

(3.3) Example. For p = 3 the equation of E can be put in the form 0 = f (w, x, y) =
2
3
3
2
2
x +awx +bw x +cw −wy . Since p−1 = 3−1 = 2, we calculate f (w, x, y) 2
2
and note that the coefficient of (wxy) is equal to −2a. Hence E is supersingular if
2
3
and only if a = 0, i.e., if it has the form y = x + bx + c. Changing x to αx + β,
3 3
3
3
we change x to α x + β and with a suitable choice of α and β the equation be-
2
3
comes y = x − x. This is the only supersingular elliptic curve in characteristic 3.
2
Moreover, E(F 3 ) ={∞= 0(0, 0), (1, 0), (−1, 0)} and this is isomorphic to (Z/2) .
The number of points is p + 1 = 3 + 1 = 4 as in (3.2).
k
k
k
(3.4) Lemma. The coefficient of x in (x − 1) (x − λ) is
k 2
k k j
(−1) λ .
j
j=0
k k k−a a k k k−b b
Proof. Expand both (x−1) = ( )(−1) x and (x−λ) = ( )(−λ) x .
a a a b
Hence the product is given by

k k k−b k k 2k−i i
(x − 1) (x − λ) = λ (−1) x .
a b
i a+b=i
k 2 j
k
Hence the coeffiecient of x is (−1) k k ( ) λ .
j=0 j