Vol. 4, No. 6, 2010

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ISSN: 1944-7833 (e-only)
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Generalized moonshine I: Genus-zero functions

Scott Carnahan

Vol. 4 (2010), No. 6, 649–679

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.

moonshine, replicable function, Hecke operator, generalized moonshine
Received: 28 December 2008
Revised: 2 October 2009
Accepted: 5 January 2010
Published: 25 September 2010
Scott Carnahan
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
United States