1846 CHAPTER 58. ELLIPTIC FUNCTIONS
Proof: Letting a basis for the module of periods of ℘ be {1,τ} , it follows from Theo-rem 58.1.2 on Page 1818 that there exists a basis for the same module of periods, {w′1,w′2}with the property that for τ ′ = w′2/w′1∣∣τ ′∣∣≥ 1,
−12≤ Reτ
′ ≤ 12.
Since this is a basis for the same module of periods, there exists a unimodular matrix,(a bc d
)such that (
w′1w′2
)=
(a bc d
)(1τ
).
Hence,
τ′ =
w′2w′1
=c+dτ
a+bτ.
Thus τ ′ is in the interior of H. In fact, it is on the interior of Ω′∪Ω≡ Q.
0 11/2−1 −1/2
τ′
58.1.7 A Short Review And Summary
With this lemma, it is easy to extend Corollary 58.1.24. First, a simple observation andreview is a good idea. Recall that when you change the basis for the module of periods, theWeierstrass ℘ function does not change and so the set of ei used in defining λ also do notchange. Letting the new basis be {w′1,w′2} , it was shown that(
w′1w′2
)=
(a bc d
)(w1w2
)
for some unimodular transformation,(
a bc d
). Letting τ = w2/w1 and τ ′ = w′2/w′1
τ′ =
c+dτ
a+bτ≡ φ (τ)