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23 Weierstrass Elliptic and Modular FunctionsModular Functions

§23.15 Definitions

  1. §23.15(i) General Modular Functions
  2. §23.15(ii) Functions λ(τ), J(τ), η(τ)

§23.15(i) General Modular Functions

In §§23.1523.19, k and k () denote the Jacobi modulus and complementary modulus, respectively, and q=eiπτ (τ>0) denotes the nome; compare §§20.1 and 22.1. Thus

23.15.1 q=exp(πK(k)K(k)),
23.15.2 k =θ22(0,q)θ32(0,q),
k =θ42(0,q)θ32(0,q).

Also 𝒜 denotes a bilinear transformation on τ, given by

23.15.3 𝒜τ=aτ+bcτ+d,

in which a,b,c,d are integers, with

23.15.4 adbc=1.

The set of all bilinear transformations of this form is denoted by SL(2,) (Serre (1973, p. 77)).

A modular function f(τ) is a function of τ that is meromorphic in the half-plane τ>0, and has the property that for all 𝒜SL(2,), or for all 𝒜 belonging to a subgroup of SL(2,),

23.15.5 f(𝒜τ)=c𝒜(cτ+d)f(τ),

where c𝒜 is a constant depending only on 𝒜, and (the level) is an integer or half an odd integer. (Some references refer to 2 as the level). If, as a function of q, f(τ) is analytic at q=0, then f(τ) is called a modular form. If, in addition, f(τ)0 as q0, then f(τ) is called a cusp form.

§23.15(ii) Functions λ(τ), J(τ), η(τ)

Elliptic Modular Function

23.15.6 λ(τ)=θ24(0,q)θ34(0,q);

compare also (23.15.2).

Klein’s Complete Invariant

23.15.7 J(τ)=(θ28(0,q)+θ38(0,q)+θ48(0,q))354(θ1(0,q))8,

where (as in §20.2(i))

23.15.8 θ1(0,q)=θ1(z,q)/z|z=0.

Dedekind’s Eta Function (or Dedekind Modular Function)

23.15.9 η(τ)=(12θ1(0,q))1/3=eiπτ/12θ3(12π(1+τ)|3τ).

In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when τ lies on the positive imaginary axis the cube root is real and positive. See also 27.14.12.