Number Theory: An Introduction to Mathematics

6 The Modular Function 531

6 The Modular Function

The function

λ(τ):=θ 104 ( 0 ;τ)/θ 004 ( 0 ;τ),

which was introduced in§5, is known as themodular function. In this section we study
its remarkable properties. (The term ‘modular function’, without the definite article, is
also used in a more general sense, which we do not consider here.)
The modular function is holomorphic in the upper half-planeH.Furthermore,we
have

Proposition 12For anyτ∈H,

λ(τ+ 1 )=λ(τ)/[λ(τ)−1],

λ(− 1 /τ)= 1 −λ(τ),

λ(− 1 /(τ+ 1 ))= 1 /[1−λ(τ)],

λ((τ− 1 )/τ)=[λ(τ)−1]/λ(τ),

λ(τ/(τ+ 1 ))= 1 /λ(τ).

Proof The first two relations have already been established in Propositions 9 and 10.
If, as in§1, we put

Uλ= 1 −λ, Vλ= 1 /( 1 −λ),

and if we also putTτ =τ+1,Sτ =− 1 /τ, then they may be written in the
form

λ(Tτ)=UVλ(τ), λ(Sτ)=Uλ(τ).

It follows that

λ(− 1 /(τ+ 1 ))=λ(STτ)=Uλ(Tτ)=U^2 Vλ(τ)=Vλ(τ)= 1 /[1−λ(τ)].

Similarly,

λ((τ− 1 )/τ)=λ(TSτ)=V^2 λ(τ)=[λ(τ)−1]/λ(τ),
λ(τ/(τ+ 1 ))=λ(TSTτ)=UV^2 λ(τ)= 1 /λ(τ).

As we saw in Proposition IV.12, together the transformationsSτ =− 1 /τand
Tτ=τ+1 generate themodular groupΓ, consisting of all linear fractional transfor-
mations

τ′=(aτ+b)/(cτ+d),

wherea,b,c,d∈Zandad−bc=1. Consequently we can deduce the effect onλ(τ)
of any modular transformation onτ. However, Proposition 12 contains the only cases
which we require.
We will now study in some detail the behaviour of the modular function in the
upper half-plane. We first observe that we need only consider the behaviour ofλ(τ)
in the right half ofH. For, from the definitions of the theta functions as infinite
series,

θ 00 ( 0 ;τ)=θ 00 ( 0 ;− ̄τ), θ 01 ( 0 ;τ)=θ 01 ( 0 ;− ̄τ),