Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 272

Since ω 1 and ω 2 are linearly independent over R ,thenumber τ cannot be
real. By switching ω 1 and ω 2 if necessary, we may assume that the imaginary
part of τ is positive:

( τ ) > 0 .

In other words, we assume τ lies in the upper half plane

H = {x + iy ∈ C | y> 0 }.

The lattice

L τ = Z τ + Z

is homothetic to L . This means that there exists a nonzero complex number
λ such that L = λL τ . In our case, λ = ω 2 .
For integers k ≥ 3, define

1
( mτ + n ) k .

G k ( τ )= G k ( L τ )= (9.11)

( m,n ) =(0 , 0)

We have

G k ( τ )= ω 2 G k ( L ) ,

where G k ( L ) is the Eisenstein series defined for L = Z ω 1 + Z ω 2 by (9.4). Let

q = e 2 πiτ .

It will be useful to express certain functions as sums of powers of q .If τ =
x + iy with y> 0, then = e − 2 πy < 1. This implies that the expressions we | q |

obtain will converge.

PROPOSITION 9.11

Let ζ ( x )= n =1 n −x and let

σ ( n )=
d

d

| n

be the sum of the thpowersofthe positive divisors of n .If k

≥ 2 isan

integer, then

∞

G 2 k ( τ )=2 ζ (2 k )+2 (2 πi ) 2 k σ 2 k− 1 ( n ) q n

(2 k − 1)!

n =1

∞

=2 ζ (2 k )+2 (2 πi ) 2 k
(2 k − 1)!

j 2 k− 1 q j
1 − q j

.

j =1