curve. We can think of Has a topological space. Since we have a quotient topology, this curve inherits some complex structure fromH. We define the genus of G as the genus of G\H. The group G=SL(2, Z) can be interpreted as the modular group of the torus. Of the groups we are interested in, most of them are commensurable withSL(2, Z). This means that G∩SL(2, Z) has finite index in both Gand SL(2, Z). The index of a subgroup G0 of a group Gis defined as |G| |G0| , which is always an integer by Lagrange’s Theorem. One family of subgroups of SL(2, Z) we are interested in is Γ0(N) :=( a b c d!∈ SL(2, Z)|N divides c) where N will determine the genus [7]. This leads us to two definitions, from the article “Monstrous Moonshine: The First Twenty-Five Years”, that will be influential in our discussion of moonshine. Definition 1: “We call a discrete subgroup Gof SL(2, R) a moonshine-type modular group if it contains some Γ0(N), and also obeys the condition that 1 t 0 1!∈ Gif and only if t ∈Z” [7]. A few notes about moonshine modular groups: first, they are commensurable with SL(2, Z). Second, if we take any meromorphic functionf : G\H→C, where Gis a moonshine modular group, it will have a Fourier expansion. We will discuss Fourier expansions later. For now, we simply note that this expansion will be of the formf(τ) = ∞ X n=−∞ anq n, where an ∈Cand q =e2πiτ [7]. Ameromorphic function is a function that is complex differentiable on all but a discrete subset of its domain, and those points must be poles. The second definition we are interested in points us to the J-function. Definition 2: “Let G be any subgroup of SL(2, R) commensurable with SL(2, Z). By a modular function f for G we mean a meromorphic function f : H→C, such that f aτ +b cτ +d =f(τ) ∀ a b c d!∈ G and such that, for any A∈SL(2, Z), the function f(A.τ) has Fourier expansion of the form ∞ X n=−∞ bnq n/N for some N and b n (both depending on A), and where bn = 0 for all but finitely many negative n”[7]. Page 42 Riley • An Overview of Monstrous Moonshine
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