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§1. Jacobi q-Parametrization: Application to Real Curves 191

which for q = e 2πiτ becomes, by (1.3),

(2πi) 2k 2k−1 mn
2ζ(2k) + 2 n q
(2k − 1)!
m,n<1
s s

where ζ(s) = (1/n ) is the Riemann zeta function. For σ s (n) = d this
1≤n d|n
formula takes the more precise form, called the q-expansion of the Eisenstein series,
(2πi) 2k n
G 2k (τ) = 2ζ(2k) + 2 σ 2k−1 (n)q
(2k − 1)!
1≤n
(2πi) 2k n 2k−1 n
q
= 2ζ(2k) + 2 .
(2k − 1)! 1 − q n
1≤n
We introduce E 2k (τ) by G 2k (τ) = 2ζ(2k)E 2k (τ).The cases2k = 4and 6are
considered in greater detail using
π 4 π 4 π 6 π 6
ζ(4) = = 2 and ζ(6) = = 3 ,
90 2 · 3 · 5 945 3 · 5 · 7
reference J.-P. Serre, Course in Arithmetic, p. 91. We have

π 4 2(2πi) 4 n
G 4 (τ) = + σ 3 (n)q
45 3!
1≤n
, 4
4
(2πi) n (2πi)
= 1 + 240 σ 3 (n)q = E 4 (τ)
4
2 · 45 720
1≤n
4
hence, g 2 (τ) = 60G 4 (τ) = [(2πi) /12]E 4 (τ),and also
2π 6 2(2πi) 6 n
G 6 (τ) = + σ s (n)q
945 5!
1≤n
, 6
6
(2πi) · 2 n −(2πi) · 2
=− 1 − 504 σ 5 (n)q = E 6 (τ)
6 3
6
2 · 945 2 3 5 · 7
1≤n
3 3
6
hence, g 3 (τ) = 140G 6 (τ) =−[(2πi) /2 3 ]E 6 (τ).
(1.5) q-Expansions of Elliptic Functions. For q = e 2πiτ and w = e 2πiτ we have
the following expansions using (1.4):
1 1 1
℘(z,τ) = + −
z 2 (z − mτ − n) 2 mτ + n) 2
(m,n) =(0,0)
1 1 1 1

= 2 − 2 2 + 2 − 2
(z − n) n (z − mτ − n) (mτ + n)
n 1≤n m =0 n