4 The J-Function We are going to work our way to the J-function, which is a modular function that connects number theory to the Monster. Modular functions and the closely related but slightly more complicated modular forms come up in number theory in a variety of ways. It was the theory of modular forms that allowed Andrew Wiles to prove Fermat’s Last Theorem. To truly understand the J-function we need to lay quite a bit of groundwork first. 4.1 Preliminary Notation and Functions We are going to need to understand elliptic functions. A function f is elliptic if it is doubly periodic and meromorphic (which means it is complex differentiable on every point of an open subset of the complex plane except for a set of isolated points called poles). Poles are isolated singular points [2]. A good way to recognize poles is where the denominator of the function is zero. A pole of order n in a rational function is a zero of the denominator of multiplicity n (which is not a zero of the numerator). A function is periodic if there is some period ω such that f(z +ω) =f(z), for z and z +ω in the domain of f. An example of a periodic function is cos 2x, which has a period of π. Graph of y = cos 2x If ω is a period of a function, so is nω for all n ∈ Z. For a function to be doubly periodic it must have two periods, ω1 and ω2 in C, with ratio ω2/ω1, where this ratio is not real. Hence, the periods will go in two different directions. An example of a function that has two periods in two different directions is cos 2x· cos 3y. It is not technically doubly periodic since the ratio of the periods is real, but it does have periods in two different directions. Doubly periodic functions can only be seen in four dimensions. Channels • 2022 • Volume 6 • Number 2 Page 35
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