Factorization of holomorphic eta quotients

Abstract

We conjecture the occurrence of a certain type of factor of a holomorphic eta quotient whenever it is reducible and we prove this conjecture for all prime power levels. In particular, this also implies that rescaling and Atkin-Lehner involutions of irreducible holomorphic eta quotients of prime power levels are irreducible. We also show that there are finitely many simple holomorphic eta quotients of a given level and provide a bound on the weights of such eta quotients of a given level. Finally, we construct an infinite family of irreducible holomorphic eta quotients of prime power levels.