Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 324

S N ,then SM has determinant N , so there exists A S ∈
SL 2 ( Z ) and a uniquely determined M S ∈

PROOF If S ∈

S N such that A S M S = SM . f

S 1 ,S 2 ∈ S N and M S 1 = M S 2 ,then

A − 1

S 1 S 1 M = M S 1 = M S 2 = A − 1 S 2 S 2 M,

which implies that A S 2 A − 1

S 1 S 1 = S 2 . By the uniqueness part of Lemma 10.10,
S 1 = S 2 . Therefore, the map S → M S is an injection on the finite set S N ,
hence is a permutation of the set. Since j ◦ A = j for A ∈ SL 2 ( Z ), we have

F N ( X, Mτ )=
S

( X − j ( SMτ ))

∈ S N

=
S

( X − j ( A S M S τ ))

∈ S N

=
S

( X − j ( M S τ ))

∈ S N

=
S

( X − j ( Sτ ))

∈ S N

= F N ( X, τ ) .

The next to last equality expresses the fact that S M S is a permutation of

S N , hence does not change the product over all of S N .
Since F N is invariant under τ → Mτ , the same must hold for its coecients
a k ( τ ).

→

LEMMA 10.12
For each k ,there existsaninteger n su ch that

q −n Z [[ q ]] , a k ( τ ) ∈

where Z [[ q ]] denotes pow er series in q withinteger coe cients. In other w ords,
a k ( τ ) can be expressed as a Laurent series w ithon y finitelymanynegative
term s, and the coe cientsareintegers.

PROOF The j -function has the expansion

= ∞

j ( τ )= 1

c k q k = P ( q ) ,

q + 744 + 196884 q +

···

k = − 1

where the coe cients c k are integers (see Exercise 9.1). Therefore,

j (( aτ + b ) /d )= ∞ c k ( ζ b e 2 πiaτ/d ) k = P ( ζ b e 2 πiaτ/d ) ,

k = − 1