Page 285 -

262 13. Elliptic Curves over Finite Fields

The function G p (λ) is a solution to the hypergeometric differential equation

2
d w dw 1
λ(1 − λ) + (1 − 2λ) + w = 0.
dλ 2 dλ 4
This can be seen by direct calculation or by using the assertion that all solutions of
p
the differential equation in F p [[λ]] are of the form F(1/2, 1/2, 1; λ)u(λ ), where
u is a formal series. Since the polynomial G p (λ) is the solution of a second-order
differential equation it follows from formal considerations that its roots are simple.
This follows from the corresponding recurrence formula for the coefficients of a
Taylor expansion around a root. Hence we have the following result.

m 2 i
(3.7) Proposition. The Deuring polynomial G p (λ) = (−1) m ( ) λ has m sim-
i
ple roots in the algebraic closure of F p .
The difference between the regular and nonregular solutions of the hypergeomet-
m
ric differential equation is only a question of a sign since G p (λ) = (−1) G p (1−λ).
There is only one period up to a constant modulo p and it is a scalar multipole of
G p (λ). In the supersingular case there are no nonzero periods. We will come back to
this when we consider the various formulations of the notion of supersingular curve.
In later sections, using the theory of isogenies and formal groups, we will have
as many as ten criterions for a curve to be supersingular. One of these can be derived
now using the relation to the Deuring polynomial.

(3.8) Proposition. An elliptic curve E is supersingular if and only if the invariant
differential ω is exact.

2
Proof. For p = 2 we see that ω = dx/(αx +1) when E has the form y +y+αxy =
3
x . Then ω is exact if and only if α = 0, i.e., E is supersingular.
2
For p > 2wehave ω = dx/2y, and we put E into the form y = x(x−1)(x−λ).
p
Then ω is exact if and only if y p−1 (dx/y ) or equivalently
p−1 m
y dx = [x(x − 1)(x − λ)] dx
is exact for m = (p−1)/2. This form is exact if and only if the coefficient of x p−1 in
m
m
[x(x − 1)(x − λ)] is zero, but this is also the coefficient of x in [(x − 1)(x − λ)] m
is zero. This coefficient is G p (λ) by (3.4). Now the proposition follows from (3.5).
These considerations can be used to count mod p the number of points on the
m
elliptic curve E λ for λ ∈ F p .For m = (p − 1)/2 recall that [x(x − 1)(x − λ)] =
2
+1or −1, an dx(x − 1)(x − λ) is a square, that is, of the form y , if and only if
m
[x(x − 1)(x − λ)] =+1. This leads to the formula
m
#E λ (F p ) = {1 + [x(x − 1)(x − λ)] } mod p.
x∈F p
Separating out the contribution form x = 0, and using the elementary character sum
in F p .