Cryptography Reference
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Continuing in this way, we obtain
g ( τ )= f ( τ ) − b n j n
−···b 0 ∈ q Z [[ q ]]
for integers b n ,...,b 0 . The function g ( τ ) is analytic in H and vanishes at i∞ .
Also, g ( τ ) is invariant under the action of SL 2 ( Z ). Proposition 9.16 says that
if g is not identically zero then a sum of the orders of g at various points is 0.
But these orders are all nonnegative since g is analytic. Moreover, the order
of g at i
is positive. Therefore the sum of the orders must be positive, hence
cannot be zero. The only possibility is that g is identically zero. This means
that
b n j n
g ( τ )= f ( τ )
−···
b 0 =0 ,
so f ( τ )
Z [ j ].
Combining Lemma 10.12 and Proposition 10.14, we obtain the first part of
the following.
THEOREM 10.15
Let N be a positive integer.
1. T here isapolynom ialw ithinteger coe cients
Φ N ( X, Y ) Z [ X, Y ]
su ch that the coe cient of the highest pow er of X is1andsuchthat
F N ( X, τ )=Φ N ( X, j ( τ )) .
2. If N is not a perfect square, then
H N ( X )=Φ N ( X, X ) Z [ X ]
isnonconstant and the coe cient of itshighest pow er of X is ± 1 .
PROOF
We have already proved the first part. For the second part, we
know that
H N ( j )=Φ N ( j, j )= F N ( j, τ )=
S
( j − j ◦ S )
S N
is a polynomial in j with integer coecients. We need to look at the coecient
of the highest power of j .Let S = ab
∈ S N . Ifweexpandthefactor
0 d
S as a Laurent series in e 2 πiτ/N , the first term for j is
j
j
e 2 πiτ =( e 2 πiτ/N ) N
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