if f(cx) =cnf(x) for some n. So g2(cω1, cω2) = 60 X mω1+nω26=0 1 (cmω1 +cnω2) 4 = 60 X mω1+nω26=0 1 c4(mω1 +nω2) 4 = 60 c4 X mω1+nω26=0 1 (mω1 +nω2) 4 = c−4g2(ω1, ω2). Similarly, g3(cω1, cω2) =c−6g3(ω1, ω2) and ∆(cω1, cω2) =c−12∆(ω1, ω2). If we take τ =ω2/ω1 and factor an ω−1 1 out of these functions, we see that g2(1, τ) = 60 X m+nτ6=0 1 (m+nτ)4 = 60 X m+n ω2 ω16 =0 1 (m+nω2 ω1 )4 = 60 X mω1+nω26=0 1 ω−4 1 (mω1 +nω2) 4 = 60 X mω1+nω26=0 ω4 1 (mω1 +nω2) 4 = ω4 1g2(ω1, ω2). Similarly, g3(1, τ) = ω6 1g3(ω1, ω2) and ∆(1, τ) = ω12 1 ∆(ω1, ω2). If τ ∈ H, we denote these three functions by g2(τ), g3(τ), and ∆(τ), respectively, and ∆(τ) = (g2(τ)) 3−27(g3(τ)) 2 6= 0 [1]. 4.2 Definition of the J-Function As we are about to define the J-function, let us survey for a moment the previous discussion. In trying to build a nonconstant elliptic function of order 2, we created the Weierstrass function. From there, we defined a relationship between Eisenstein series and the coefficients of a differential equation that ℘(z) satisfies. This relationship gave us g2 and g3 which we then used to define ∆. We defined all three of those functions in terms of τ. We now define Klein’s function. This function is a quotient of g2 and ∆. A function of the periods ω1 and ω2, it is homogeneous of degree 0. This means that we can multiply ω1 and ω2 by a constant and not change the value of the function. For ω2/ω1 6∈R, we define Page 40 Riley • An Overview of Monstrous Moonshine
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