Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 326

Since ad = N for each matrix in S N ,

e 2 πiaτ/d = e 2 πia 2 τ/N .

Therefore, the p k ( τ ) in the claim can be regarded as a Laurent series in
e 2 πiτ/N . The claim implies that the coecients a k ( τ )of F N ( X, τ ) are Laurent
series in e 2 πiτ/N with integer coecients. To prove the lemma, we need to
remove the N .Thematrix

11
01

∈ SL 2 ( Z )

acts on τ + 1. Lemma 10.11 implies that a k ( τ ) is invariant under

τ → τ +1. Since ( e 2 πiτ/N ) is invariant under τ → τ +1only when N | ,the
Laurent series for a k must be a Laurent series in ( e 2 πiτ/N ) N

H by τ →

= e 2 πiτ .This

proves Lemma 10.12.

PROPOSITION 10.14
Let f ( τ ) be analyticfor τ ∈H , and suppose
f aτ + b
cτ + d

= f ( τ )

for all ab

∈ SL 2 ( Z ) and all τ ∈H .Also, assu m e

cd

f ( τ ) ∈ q −n Z [[ q ]]

forsomeinteger n .Then f ( τ ) isapolynom ialin j withinteger coe cients:

f ( τ ) ∈ Z [ j ] .

PROOF

Recall that

1
q ∈ Z [[ q ]] .

j ( τ ) −

Write

f ( τ )= b n

q n +

··· ,

with b n ∈ Z .Then

b n − 1

f ( τ ) − b n j n =

q n− 1 + ··· ,

with b n− 1 ∈ Z . Therefore,

b n − 2

b n j n b n− 1 j n− 1 = f ( τ ) − −

q n− 2 +

··· .