Holomorphic modular functions on moonshine groups
University of Bristol
For any square-free integer N such that "moonshine group" Gamma0(N)^+ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of Gamma0(N)^+ to certain McKay-Thompson series associated to the representation theory of the Fischer-Griess monster group. In particular, the Hauptmoduli admits a q-expansion which has integer coefficients. In joint work with Jay Jorgenson and Lejla Smajlovic, we study the holomorphic function theory associated to higher genus moonshine groups Gamma0(N)^+. For all genus one groups, we prove that the corresponding function field admits two generators whose q-expansions have integer coefficients. As a corollary, we derive the cubic relation which defines the underlying elliptic curve.