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Computational aspects of Monstrous moonshine

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Monstrous moonshine is a surprising and mysterious connection between the monster group M (and also other finite sporadic simple groups) and modular functions. Its research was initiated thirty years ago by John Conway, John McKay and Simon Norton, and continued by an always growing community of mathematicians, including the Fields Medal winner Richard E. Borcherds, who proved the original conjectures. Although in the last years much has been discovered, there is still a lot to work on. I am interested in several computational aspects of this theory, as well as in learning the theoretical background, which is extensive to say the least, given that this is a multidisciplinary topic.

The starting point of Monstrous moonshine is the observation that there is a relation between the coefficients of the Fourier expansion of the classical function j,

j(q) = 1/q + 744 + 196884 q + 21493760 q2 + 864299970 q3 + ⋯ , with q = e2πiz

and the dimensions of the irreducible representations of M, the biggest sporadic simple group ("the Monster"),

1, 196883, 21296786, 842609326, 18538750076, …

In fact, there are 171 functions that are related to the 172 conjugacy classes of M. All these functions are generators for the fields of functions (subfields of C(x)) left invariant by certain genus-0 groups. Some important properties of these can be generalized, leading to a larger set of 619 called replicable functions. At the time of my involvement, there was no definitive proof that the 619 functions that had been calculated were the only existing ones, but only some independent checks.

In relation to this, I used my tools in functional decomposition to find rational relations among these functions and refine them. An advance of this is in [2] and the complete results in [1]. It may be possible for us to exploit some of these relations with the purpose of relating the expressions of moonshine and replicable functions in terms of more classical functions, like η-products and Eisenstein functions. On the other hand, a process of graph refinement did not produce any new replicable functions, yet another piece of evidence in support of the completeness of the list.

Another problem is to compute the Galois groups of the function fields generated by the critical values given by replicable functions; we expect them to be dihedral or similar, this will provide interesting information from a number-theoretical point of view.

Finally, yet another computation that may yield some new information is that of the characters of the inverse functions. We expect to be able to generate some general formulas in this case, which could provide an alternative point of view for the expressions of replicable functions.

Related publications

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  1. Decomposing replicable functions (with J. McKay), LMS Journal of Computation and Mathematics 11 (June 2008), p. 146-171. ISSN 1461-1570. DOI 10.1112/S1461157000000553. Download PDF
  2. Aplicación de la descomposición racional univariada a Monstrous Moonshine In Spanish (with J. McKay), Proceedings of the 2004 Encuentro de Álgebra Computacional y Aplicaciones (EACA 2004), p. 289-294. ISBN 84-688-6988-04. Download PDF