Generalized Moonshine I: Genus zero functions

TL;DR

This paper introduces a new notion of Hecke-monicity for functions related to elliptic curves and demonstrates that such functions with algebraic integer coefficients are either genus-zero functions invariant under congruence groups or of a degenerate type, extending moonshine theory.

Contribution

It defines Hecke-monicity for functions on moduli spaces, proves that weakly Hecke-monic functions with certain properties are genus-zero or degenerate, and applies this to characters of Lie algebras from conformal field theory.

Findings

Weakly Hecke-monic functions with algebraic integer coefficients are genus-zero or degenerate.

Replicable functions of finite order are shown to be genus-zero functions.

Characters of certain Lie algebras are proven to be weakly Hecke-monic and genus-zero under specific conditions.

Abstract

We introduce a notion of Hecke-monicity for functions on certain moduli spaces associated to torsors of finite groups over elliptic curves, and show that it implies strong invariance properties under linear fractional transformations. Specifically, if a weakly Hecke-monic function has algebraic integer coefficients and a pole at infinity, then it is either a holomorphic genus-zero function invariant under a congruence group or of a certain degenerate type. As a special case, we prove the same conclusion for replicable functions of finite order, which were introduced by Conway and Norton in the context of monstrous moonshine. As an application, we introduce a class of Lie algebras with group actions, and show that the characters derived from them are weakly Hecke-monic. When the Lie algebras come from chiral conformal field theory in a certain sense, then the characters form holomorphic genus-zero functions invariant under a congruence group.