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23 Weierstrass Elliptic and Modular FunctionsModular Functions

§23.18 Modular Transformations

Elliptic Modular Function

λ(𝒜τ) equals

23.18.1 λ(τ),

according as the elements [abcd] of 𝒜 in (23.15.3) have the respective forms

23.18.2 [oeeo],

Here e and o are generic symbols for even and odd integers, respectively. In particular, if a-1,b,c, and d-1 are all even, then

23.18.3 λ(𝒜τ)=λ(τ),

and λ(τ) is a cusp form of level zero for the corresponding subgroup of SL(2,).

Klein’s Complete Invariant

23.18.4 J(𝒜τ)=J(τ).

J(τ) is a modular form of level zero for SL(2,).

Dedekind’s Eta Function

23.18.5 η(𝒜τ)=ε(𝒜)(-i(cτ+d))1/2η(τ),

where the square root has its principal value and

23.18.6 ε(𝒜)=exp(πi(a+d12c+s(-d,c))),
23.18.7 s(d,c)=r=1(r,c)=1c-1rc(drc-drc-12),

Here the notation (r,c)=1 means that the sum is confined to those values of r that are relatively prime to c. See §27.14(iii) and Apostol (1990, pp. 48 and 51–53). Note that η(τ) is of level 12.