Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 308

τ ).
(b) Show that if τ is in the fundamental domain

(a) Let τ ∈F . Show that j ( τ )= j ( −

F , then either −τ ∈F ,

or ( τ )= − 1 / 2, or |τ | =1with − 1 / 2 ≤ ( τ ) ≤ 0.
(c) Suppose that τ ∈F and that j ( τ ) ∈ R . Show that ( τ )=0,
or ( τ )= − 1 / 2, or | τ | =1with − 1 / 2 ≤ ( τ ) ≤ 0( Hint: Use

Corollary 9.18.)
(d) Let τ ∈H . Show that if |τ | =1then ( − 1 / ( τ +1))= − 1 / 2.
(e) Let L be a lattice with g 2 ( L )= −A and g 3 ( L )= −B .

Show

that there exists τ ∈H ( τ )=0or

such that − 1 / 2and j ( L )=

j ( Z τ + z ).
(f) Show that if τ ∈H is such that ( τ )=0or − 1 / 2, then we have
g 2 ( τ ) ,g 3 ( τ ) ∈ R .
(g) By Corollary 9.20, there exists λ

C such that L =( λ )( Z τ + Z ). ∈

=0 , 1728 then λ 2 Show that if j ∈ R . Hint: Use Equations

(9.14).)
This shows that L is obtained from the lattice Z τ + Z by an ex-
pansion by and a rotation by 0 , 90 ◦ , 180 ◦ , or 270 ◦ .

(h) Let 0 = y ∈ R .Let M be the lattice ( 2 + iy ) Z + Z . Show that iM
has {y + 2 i, 2 y} as a basis.
(i) Assume that j

| λ |

=0 , 1728. Show that L has a basis { ω 1 ,ω 2 } with

( ω 1 )=0or 2 ω 2 . Therefore, the lattice L is either
rectangular or a special shape of parallelogram.
(j) Use the facts that j ( ρ )=0and j ( i ) = 1728 to prove (i) in the
cases that j ( E )=0and j ( E ) = 1728. (The condition that λ 2

ω 2 ∈ R and

∈ R

gets replaced by λ 6 R and λ 4 R , respectively. However, the

lattices for τ = ρ and τ = i have extra symmetries.)

∈ ∈

9.6 Let L be a lattice that is stable under complex conjugation (that is, if
ω ∈ L then ω ∈ L ). This is the same as requiring that the elliptic curve
associated to L is defined over R (see Exercise 9.5).

(a) Show that ℘ ( z )= ℘ ( z ).
(b) Show that if t ∈ R and if ω 2 ∈ R is a real period, then

℘ 1

2 ω 2 + it

∈ R .

( Hint: Use (a), the periodicity of ℘ , and the fact that ℘ ( − z )=

℘ ( z ).)
(c) Differentiate the result of (b) to show that ℘ ( z ) ∈ i R for the
points

1

2 ω 2 + it in (b). This path, for 0 ≤ t ≤ ω 1 , corresponds to
x moving along the x -axisbetweenthetwopartsofthegraphin