Klein’s function as J(ω1, ω2) = g3 2(ω1, ω2) ∆(ω1, ω2) = λ−12g3 2(ω1, ω2) λ−12∆(ω1, ω2) = g3 2(λω1, λω2) ∆(λω1, λω2) =J(λω1, λω2) for λ6= 0. This equality is true since bothg3 2 and ∆ are homogeneous of degree −12. Because of this fact, for τ ∈Hwe see that J(1, τ) =J(ω1, ω2). We will denote this function simply as J(τ) [1]. If we define ω0 2 = aω2 +bω1 and ω0 1 = cω2 +dω1, where a, b, c, d ∈ Z such that ad−bc = 1, this new pair of periods generates the same lattice Ω. Note that τ0 = ω0 2 ω0 1 = aω2 +bω1 cω2 +dω1 = aω2 +bω1 cω2 +dω1 · 1 ω1 1 ω1 = aω2 ω1 +b cω2 ω1 +d = aτ +b cτ +d . Functions of the formf(τ) = aτ+b cτ+d where a, b, c, d are integers and ad −bc = 1 are called unimodular transformations. The set of all these transformations form a group under composition called the modular group. Hence, “J(τ) is invariant under the transformations of the modular group” [1]. In other words, J aτ+b cτ+d =J(τ). One instance of this is τ0 =τ+1, which means that J(τ) is a periodic function with period of 1, since both τ and τ0 generate the same lattice. Let us pause our discussion of the J-function temporarily to show that SL(2, Z) behaves in this way. This is important for our focus on monstrous moonshine. 4.3 Modular Functions in General To begin, we will consider the group SL(2, R) of 2 ×2 matrices with entries fromR with determinant 1. We will let SL(2, R) act on the upper half plane H={τ ∈C|Im(τ) >0} by a b c d!· τ = aτ +b cτ +d [7]. Technically we are dealing with SL(2, R)/{±I} on H. In other words, the quotient group of SL(2, R) modded out by {±I}, since ±1 0 0 ±1!· τ = ±τ + 0 0· τ ±1 =τ and hence I and −I act as the identity element. For the purposes of this paper we will think of it in terms of SL(2, R) [7]. If we let G be a discrete subgroup of SL(2, R), we can regard G\ H (a quotient topology where all points in one equivalence class get mapped to one point) as a complex Channels • 2022 • Volume 6 • Number 2 Page 41
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