Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 277

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Figure 9.3
The Fundamental Domain for SL 2 ( Z )

The proposition says that F is a fundamental domain for the action of

SL 2 ( Z )on H . For a proof of the proposition, see [104] or [108].

COROLLARY 9.15
Let L ⊂ C be a lattice. T here existsabasis {ω 1 ,ω 2 } of L with ω 1 /ω 2 ∈F .
In o ther w ords,

L =( λ )( Z τ + Z )

C × and som e uniquelydeterm ined τ

for som e λ

∈ ∈F .

PROOF Let {α, β} be a basis for L and let τ 0 = α/β . By changing the
sign of α if necessary, we may assume that τ 0 ∈H .Let
ab
cd

∈ SL 2 ( Z )

be such that

aτ 0 + b
cτ 0 + d

= τ ∈F.

Let

ω 1 = aα + bβ, ω 2 = cα + dβ.

Since the matrix is in SL 2 ( Z ),

L = Z α + Z β = Z ω 1 + Z ω 2 = ω 2 ( Z τ + Z ) .

This proves the corollary.