1848 CHAPTER 58. ELLIPTIC FUNCTIONS

Corollary 58.1.26 λ′ (τ) ̸= 0 for all τ in the upper half plane, denoted by P+.

Proof: Let τ ∈ P+. By Lemma 58.1.25 there exists φ a unimodular transformation andτ ′ in the interior of Q such that τ ′ = φ (τ). Now from the definition of λ in terms of the ei,there is at worst a permutation of the ei and so it might be the case that λ (φ (τ)) ̸= λ (τ)but it is the case that λ (φ (τ)) = ξ (λ (τ)) where ξ

′ (z) ̸= 0. Here ξ is one of the functionsdetermined by 58.1.34 - 58.1.39. (Since λ (τ) /∈ {0,1} , ξ

′ (λ (z)) ̸= 0. This follows fromthe above possibilities for ξ listed above in 58.1.34 - 58.1.39.) All the possibilities areξ (z) =

z,z−1

z,

11− z

,1− z,z

z−1,

1z

and these are the same as the possibilities for ξ−1. Therefore,

λ′ (φ (τ))φ

′ (τ) = ξ′ (λ (τ))λ

′ (τ)

and so λ′ (τ) ̸= 0 as claimed.

Now I will present a lemma which is of major significance. It depends on the remark-able mapping properties of the modular function and the monodromy theorem from ana-lytic continuation. A review of the monodromy theorem will be listed here for convenience.First recall the definition of the concept of function elements and analytic continuation.

Definition 58.1.27 A function element is an ordered pair, ( f ,D) where D is an open balland f is analytic on D. ( f0,D0) and ( f1,D1) are direct continuations of each other ifD1 ∩D0 ̸= /0 and f0 = f1 on D1 ∩D0. In this case I will write ( f0,D0) ∼ ( f1,D1) . Achain is a finite sequence, of disks, {D0, · · · ,Dn} such that Di−1 ∩Di ̸= /0. If ( f0,D0) is agiven function element and there exist function elements, ( fi,Di) such that {D0, · · · ,Dn}is a chain and

(f j−1,D j−1

)∼ ( f j,D j) then ( fn,Dn) is called the analytic continuation of

( f0,D0) along the chain {D0, · · · ,Dn}. Now suppose γ is an oriented curve with parameterinterval [a,b] and there exists a chain, {D0, · · · ,Dn} such that γ∗ ⊆ ∪n

k=1Dk,γ (a) is thecenter of D0,γ (b) is the center of Dn, and there is an increasing list of numbers in [a,b] ,a=s0 < s1 · · · < sn = b such that γ ([si,si+1]) ⊆ Di and ( fn,Dn) is an analytic continuation of( f0,D0) along the chain. Then ( fn,Dn) is called an analytic continuation of ( f0,D0) alongthe curve γ . (γ will always be a continuous curve. Nothing more is needed. )

Then the main theorem is the monodromy theorem listed next, Theorem 55.4.5 and itscorollary on Page 1749.

Theorem 58.1.28 Let Ω be a simply connected subset of C and suppose ( f ,B(a,r)) is afunction element with B(a,r)⊆ Ω. Suppose also that this function element can be analyt-ically continued along every curve through a. Then there exists G analytic on Ω such thatG agrees with f on B(a,r).

Here is the lemma.

Lemma 58.1.29 Let λ be the modular function defined on P+ the upper half plane. Let Vbe a simply connected region in C and let f : V → C\{0,1} be analytic and nonconstant.Then there exists an analytic function, g : V → P+ such that λ ◦g = f .