Monstrous Moonshine over Z?

TL;DR

This paper reviews recent advances in Monstrous Moonshine, including the resolution of key conjectures, and proposes speculative ideas to extend and unify these theories in the context of modular forms and group theory.

Contribution

It introduces new speculative conjectures aiming to extend and unify the recent resolutions of Norton's and Ryba's Moonshine conjectures.

Findings

Norton's Generalized Moonshine conjecture resolved

Ryba's Modular Moonshine conjecture resolved

Proposes new conjectures for unification and extension

Abstract

Monstrous Moonshine was extended in two complementary directions during the 1980s and 1990s, giving rise to Norton's Generalized Moonshine conjecture and Ryba's Modular Moonshine conjecture. Both conjectures have been unconditionally resolved in the last few years, so we describe some speculative conjectures that may extend and unify them.