Characterizing Moonshine Functions by Vertex-Operator-Algebraic Conditions

Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of $V$ has graded trace given by a “completely replicable function”, and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces $171$ such functions, known as the monstrous moonshine functions. We show that $154$ of the $157$ non-monstrous completely replicable functions cannot possibly occur as trace functions on $V$.