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§3. Definition of Supersingular Elliptic Curves 261

2
(3.5) Proposition. In characteristic p > 2 the curve in Legendre form E λ : y =
x(x − 1)(x − λ) is supersingular if and only if λ is a root of the Hasse invariant
m 2 i p − 1
m

m
H p (λ) = (−1) λ , where m = .
i 2
i=0
Proof. We must calculate the coefficient of (wxy) p−1 in f p−1 where
2
f (w, x, y) = wy − x(x − w)(x − λw).
In f (w, x, y) p−1 the term (wxy) p−1 will appear only in the middle of the binomial
p−1 2 m m p−1
expansion, i.e., the term ( )(wy ) [x(x − w)(x − λw)] . Observe that ( ) =
m m
m
p
p
(−1) (mod p) from the relation (x + y) p−1 (x + y) = x + y (mod p). Since
m
2m = p − 1, the coefficient of (wxy) p−1 is just the coefficient of (xw) in (x −
m
m
m
w) (x − λw) up to (−1) and by Lemma (3.4) this is H p (λ) which proves the
proposition.
In Chapter 9 we saw how the hypergeometric function F(1/2, 1/2, 1; λ) played
a basic role in describing the period lattice of an elliptic curve over C. This same
function is related to the Hasse invariant H p (λ). To see this, observe that the coeffi-
cients

1 3
1 − − ...(1 − 2k/2) k

− 2 2 −1 1 · 3 · 5 ...(2k − 1)
2
= =
k k! 2 k!
k
2k
1
1
= (−1) k are in Z .
2 k 2
Hence reducing modulo p, we see that the formal series F(1/2, 1/2, 1; λ) in F p [[λ]]
is defined, and moreover we denote by
m 2 1 1
m
i p
G p (λ) = λ ≡ F , , 1; λ mod (p,λ )
i 2 2
i=0
The polynomial G p (λ) is called the Deuring polynomial. Since for m = (p − 1)/2,
we have the congruence
m 1 (p − 1)(p − 3)...(p − 2k + 1) −1 1 · 3 · 5 ··· (2k − 1)

k k
= ≡
k 2 k! 2 k!

1
− 2
= (mod p) for k < p.
k
(3.6) Remark. In the ring of formal series F p [[λ]] the following relation holds:
1 1 p p 2

F , , 1; λ = G p (λ) · G p (λ ) · G p (λ )... .
2 2