Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 454

Figure 15.1
A Fundamental Domain for Γ 0 (2)

is a cyclic subgroup of E τ of order N .Let
γ = ab
cd

∈ Γ 0 ( N )

and let

γτ = aτ + b

cτ + d .

Since

Z τ + Z = Z ( aτ + b )+ Z ( cτ + d )=( cτ + d )( Z γτ + Z ) ,

there is an isomorphism

f γ : C / ( Z τ + Z ) −→ C / ( Z γτ + Z )

given by

f γ ( z )= z/ ( cτ + d ) .

This isomorphism between E τ and E γτ maps the point k/N to

k
N ( cτ + d )

ka

N − k c

aτ + b
cτ + d

=

N

ka
N

≡ mod Z γτ + Z

(wehaveusedthefactthat c ≡ 0(mod N )). Therefore, the subgroup C τ
of E τ is mapped to the corresponding subgroup C γτ of E γτ ,so f γ maps the
pair ( E τ ,C τ ) to the pair ( E γτ ,C γτ ). We conclude that if τ 1 ,τ 2 ∈H are
equivalent under the action of Γ 0 ( N ), then the corresponding pairs ( E τ j ,C τ j )
are isomorphic. It is not hard to show that, conversely, if the pairs are iso-
morphic then the corresponding τ j 's are equivalent under Γ 0 ( N ). Moreover,