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190 10. Theta Functions

observe that L τ = L τ+k for any integer k. Hence the exponential e 2πiτ = q or q τ ,
has importance for elliptic functions associated with the lattice L τ .
∗
(1.2) Remark. Under the exponential map z → e 2πiz = w defined C → C with
Z
∗
kernel Z, the lattice L = Zτ + Z in C is mapped to q in C , the infinite cyclic
subgroup generated by q = e 2πiτ with |q| < 1. Hence, by passing to the quotient
Z
torus, we have an analytic isomorphism E τ = C/L τ → C /q .
∗
Now we study the form of the Weierstrass equation and the elliptic functions in
terms of w = e 2πiz and q = e 2πiτ . These expressions in the new parameters are
derived by using the following expansion from complex analysis
1 π 2
= .
2
(ζ + n) 2 sin πζ
n∈Z
For T = e 2πiζ this becomes
1 1 e 2πiζ
2 2
= (2πi) = (2πi)
(ζ + n) 2 (e inζ − e −inζ 2 (1 − e 2πiζ 2
)
)
n∈Z
T 2 n
2
= (2πi) = (2πi) nT .
(1 − T ) 2
1≤n
From the chain rule we have d/dζ = 2πiT (d/dT ) and hence (d/dζ) k−2 =
(2πi) k−2 (T (d/dT ) k−2 . Applying this to the previous relation, we obtain the fol-
lowing power series relation.
(1.3) Remark. For T = e 2πiζ we have the follwing expansion related to the series
k−1 n
g k (T ) = n T :
1≤n
1 k k−1 n k
k

(k − 1)!(−1) = (2πi) n T = (2πi) g k (T ).
(ζ + n) k
n 1≤n
k
Note that g k+1 (T ) = (T [d/dT ])g k (T ) and g k (1/T ) = (−1) g k (T ). Two important
special cases used later are
T 1 + T
g 2 (T ) = 2 and g 3 (T ) = T 3 .
(1 − T ) (1 − T )
(1.4) q-Expansion of Eisenstein Series. The following modular functions, called
Eisenstein series, arose previously as coefficients of the differential equation for the
Weierstrass ω-function

1
G 2k (τ) =
(mτ + n) 2k
(m,n) =(0,0)
1 1
= 2 + 2
n 2k (mτ + n) 2k
1≤n 1≤m n