Elliptic Curves over Finite Fields - Elliptic Curves: Number Theory and Cryptography - page 136

Using the fact that
b k +1 = ( p

2

1) / 2
k +1

−

=

2

(( p − 1) / 2)!

( k + 1)!((( p − 1) / 2) − k − 1)!

= (( p − 1) / 2) − k
k +1

2

b k ,

we find that the coecient of T k

is

4((( p − 1) / 2) − k ) 2

− (2 k +1) 2 b k = p ( p − 2 − 4 k ) b k ≡ 0(mod p ) .

This proves the claim.
Suppose now that H p ( λ )=0with λ =0,

we have λ =0 , 1. Write H p ( T )=( T−λ ) r G ( T ) for some polynomial G ( T )with
G ( λ ) = 0. Suppose r ≥ 2. In (4.5), we have ( T − λ ) r− 1 dividing the last term
and the middle term, but only ( T − λ ) r− 2 divides the term 4 T (1 − T ) H p ( T ).
Since the sum of the three terms is 0, this is impossible, so we must have
r = 1. Therefore, λ is a simple root. ( Technical point: Since the degree of
H p ( T ) is less than p ,wehave r<p , so the first term of the derivative

∈ F p .Since H p (0) =0and H p (1)

H p ( T )= r ( r λ ) r− 2 G ( T )+2 r ( T λ ) r− 1 G ( T )+( T λ ) r G ( T ) − 1)( T − − −

does not disappear in characteristic p . Hence ( T − λ ) r− 1 does not divide the
first term of (4.5).)

REMARK 4.39 The differential equation 4.5 is called a Picard-Fuchs
differential equation . For a discussion of this equation in the study of
families of elliptic curves in characteristic 0, see [24]. Once we know that
H p ( T ) satisfies this differential equation, the simplicity of the roots follows
from a characteristic p version of the uniqueness theorem for second order
differential equations. If λ is a multiple root of H p ( T ), then H p ( λ )= H p ( λ )=
0. Such a uniqueness theorem would say that H p ( T ) must be identically 0,
which is a contradiction. Note that we must avoid λ =0 , 1 because of the
coe cient T (1 − T )for H p ( T ).

COROLLARY 4.40
Let p ≥ 5 be prime.Thenumberof j ∈ F p that occur as j -invariantsof
su persingular elliptic curves is

p

12 + p ,

where p =0 , 1 , 1 , 2 if p ≡ 1 , 5 , 7 , 11 (mod 12) , respectively.