1 Introduction In 1979, John Conway and Simon Norton published a paper conjecturing a connection between the Monster group and the J-invariant function. It was this paper that first introduced the term “moonshine”, or crazy idea, in connection with this conjecture [11]. The name stuck, and the title of “monstrous moonshine” was here to stay. Even before Griess constructed the Monster, amazing discoveries were already being discussed. These were often tied to a conjectured 196 883-dimensional irreducible representation of the Monster. When Griess constructed this representation, he added new mysteries, such as a new algebra which he used to construct the Monster. Group theorists wanted to know why the dimensions of the irreducible representations of the Monster were tied to the coefficients of the J-function. This connection between group theory and number theory took a collection of brilliant minds to discover. After Conway and Norton conjectured the connection, Frenkel, Lepowsky, and Meurman helped the problem by diving into the area of vertex operator algebras. Vertex operator algebras were being used in string theory, so that connection was unexpected but intriguing. Borcherds then built on the work of Frenkel, Lepowksy, and Meurman to solve the conjecture. Although the original conjecture has a solution, there are still unanswered questions related to group theory, modular forms, and physics. In this paper, we will work through some key elements of the monstrous moonshine conjecture and solution. We will start with an overview of group theory that will give us enough background information to understand the Monster group. After that, we will briefly discuss character theory, which will allow us to comprehend some of the basics of the argument for the monstrous moonshine conjecture. From there, we will put the Monster on hold temporarily to discuss modular functions and specifically the J-function. There we will clearly see the connection between the Monster and the J-function that intrigued so many. We will look at some of Frenkel, Lepowsky, and Meurman’s work creating a vertex operator algebra, which will lead us to Borcherds’ solution to the monstrous moonshine problem. The monstrous moonshine conjecture is surprising but beautiful. The theory connects different fields in a fascinating way. Although there are still questions related to moonshine, the progress that has been made in the past forty years has been impressive. Seeing what has been accomplished should give us optimism about what has yet to be discovered. Channels Vol. 6 No. 2 (2022): 27–50 ISSN 2474-2651 © 2022, Catherine E. Riley, licensed under CC BY-NC-ND (http://creativecommons.org/licenses/by-nc-nd/4.0/) Channels • 2022 • Volume 6 • Number 2 Page 27
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