From the text: We survey the theory of replicable functions and matters of related interest. These functions were introduced in monstrous moonshine, characterizing the principal moduli attached to the conjugacy classes of the monster simple group, M. It is not surprising that replicable functions, being related to moonshine and the monster, have a wide range of connections to other fields of mathematics and physics which remain to be fathomed. Indeed, moonshine has been described as 21st. century mathematics in the 20th. century. Having arrived, we can survey the past 25 years with some satisfaction but there is much remaining to be clarified and put into an appropriate context. The field is amazingly fertile: there are connections with several aspects of mathematical physics and number theory, and one finds classical and modern themes continually coming into play. We explain a few of these connections, some of which are presented here for the first time.
It is simplest to define replicable functions through the Faber polynomials to which the next section is devoted. We then provide examples related to classical themes such as Chebyshev polynomials and Hecke operators. Later sections will deal with the automorphic aspect of the replicable functions, links with the Schwarz derivative, the characterization of the monstrous moonshine functions, the exceptional affine correspondences, class numbers, and the soliton equations and their \(\tau\)-function from the 2D Toda hierarchy.
For the entire collection see [Zbl 1104.11003].