Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 271

manipulation shows that this equals the value for − y 3 obtained from the ad-

dition law for ( x 1 ,y 1 )+( x 2 ,y 2 )=( x 3 , − y 3 ). Therefore,

( ℘ ( z 1 ) ,℘ ( z 1 )) + ( ℘ ( z 2 ) ,℘ ( z 2 )) = ( ℘ ( z 1 + z 2 ) ,℘ ( z 1 + z 2 )) .

This is exactly the statement that

Φ( z 1 )+Φ( z 2 )=Φ( z 1 + z 2 ) . (9.9)

It remains to check (9.9) in the cases where (9.7) is not defined. The
cases where ℘ ( z i )= ∞ and where z 1 ≡−z 2 mod L are easily checked. The
remaining case is when z 1 = z 2 .Let z 2 → z 1 in (9.7), use l'Hopital's rule, and
use (9.8) to obtain

℘ ( z 1 )
℘ ( z 1 )

2

℘ (2 z 1 )= 1 − 2 ℘ ( z 1 )

4

6 ℘ ( z 1 ) 2 2

1

−

2 g 2

1
4

= − 2 ℘ ( z 1 ) (9.10)

℘ ( z 1 )

6 x 1 − 2

1

2 g 2

1
4

= − 2 x 1 .

y 1

This is the formula for the coordinate x 3 that is obtained from the addition
law on E . Differentiating with respect to z 1 yields the correct formula for the
y -coordinate, as above. Therefore,

Φ( z 1 )+Φ( z 1 )=Φ(2 z 1 ) .

This completes the proof of the theorem.

The theorem shows that the natural group law on the torus C /L matches
the group law on the elliptic curve, which perhaps looks a little unnatural.
Also, the classical formulas (9.7) and (9.10) for the Weierstrass ℘ -function,
which look rather complicated, are now seen to be expressing the group law
for E .

9.3 Elliptic Curves over C

In the preceding section, we showed that a torus yields an elliptic curve. In
the present section, we'll show the converse, namely, that every elliptic curve
over C comes from a torus.
Let L = Z ω 1 + Z ω 2 be a lattice and let

τ = ω 1 /ω 2 .