1832 CHAPTER 58. ELLIPTIC FUNCTIONS
Observation 58.1.15 The function, λ (τ) is analytic for τ in the upper half plane and neverassumes the values 0 and 1.
This is a very interesting function. Consider what happens when(w′1w′2
)=
(a bc d
)(w1w2
)and the matrix is unimodular. By Theorem 58.1.12 on Page 1823 {w′1,w′2} is just anotherbasis for the same module of periods. Therefore, ℘(z,w1,w2) =℘(z,w′1,w
′2) because both
are defined as sums over the same values of w, just in different order which does not matterbecause of the absolute convergence of the sums on compact subsets of C. Since ℘ isunchanged, it follows ℘′ (z) is also unchanged and so the numbers, ei are also the same.However, they might be permuted in which case the function λ (τ) defined above wouldchange. What would it take for λ (τ) to not change? In other words, for which unimodulartransformations will λ be left unchanged? This happens if and only if no permuting takesplace for the ei. This occurs if ℘
(w12
)=℘
(w′12
)and ℘
(w22
)=℘
(w′22
). If
w′12− w1
2∈M,
w′22− w2
2∈M
then℘(w1
2
)=℘
(w′12
)and so e1 will be unchanged and similarly for e2 and e3. This occurs
exactly when12((a−1)w1 +bw2) ∈M,
12(cw1 +(d−1)w2) ∈M.
This happens if a and d are odd and if b and c are even. Of course the stylish way to saythis is
a≡ 1mod2, d ≡ 1mod2, b≡ 0mod2, c≡ 0mod2. (58.1.11)
This has shown that for unimodular transformations satisfying 58.1.11 λ is unchanged.Letting τ be defined as above,
τ′ =
w′2w′1≡ cw1 +dw2
aw1 +bw2=
c+dτ
a+bτ.
Thus for unimodular transformations,(
a bc d
)satisfying 58.1.11, or more succinctly,
(a bc d
)∼(
1 00 1
)mod2 (58.1.12)
it follows that
λ
(c+dτ
a+bτ
)= λ (τ) . (58.1.13)
Furthermore, this is the only way this can happen.