Finiteness of irreducible holomorphic eta quotients of a given level

TL;DR

This paper proves that for any positive level, only finitely many irreducible holomorphic eta quotients exist, and constructs such quotients for all cubefree levels, showing they can have arbitrarily large weights.

Contribution

It establishes finiteness of irreducible holomorphic eta quotients at any level and provides explicit constructions for all cubefree levels.

Findings

Finiteness of irreducible holomorphic eta quotients at each level.

Existence of irreducible holomorphic eta quotients for all cubefree levels.

Construction of such eta quotients with arbitrarily large weights.

Abstract

We show that for any positive integer $N$, there are only finitely many holomorphic eta quotients of level $N$, none of which is a product of two holomorphic eta quotients other than 1 and itself. This result is an analog of Zagier's conjecture/ Mersmann's theorem which states that: Of any given weight, there are only finitely many irreducible holomorphic eta quotients, none of which is an integral rescaling of another eta quotient. We construct such eta quotients for all cubefree levels. In particular, our construction demonstrates the existence of irreducible holomorphic eta quotients of arbitrarily large weights.