[Paper Review] Algorithmic determination of irreducibility of holomorphic eta quotients

Soumya Bhattacharya|arXiv (Cornell University)|Feb 9, 2016

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper presents an algorithmic solution to determining the irreducibility of holomorphic eta quotients by proving that if such a quotient of level $N$ has no nontrivial factors of level dividing $N$, then all its factors have levels bounded above by an explicit function of $N$. The key result establishes that reducible eta quotients of prime power level $N$ must have a nontrivial factor whose level divides $N$, enabling irreducibility certification via level-based factorization checks.

ABSTRACT

Although eta quotients have been extensively studied for a long time,the fundamental problem of constructing an algorithm that determines whether or not a given holomorphic eta quotient $f$ is a product of two holomorphic eta quotients other than 1 and itself still remains open. The difficulty of the problem stems from an apparent lack of information about the existence of an upper bound for the levels of the factors of $f$. Here we show that if a holomorphic eta quotient $f$ of level $N$ has no nontrivial factors whose level divide $N$, then the levels of all the factors of $f$ are bounded above with respect to $N$. We also provide an explicit upper bound in terms of $N$ for the minimum of the levels of the nontrivial factors of $f$. This bound has a further refinement with respect to the weight of $f$. In particular, we show that any reducible holomorphic eta quotient of a prime power level $N$ has a nontrivial factor whose level divides $N$. As a consequence, it follows that all rescalings by positive integers and all Atkin-Lehner involutions of irreducible holomorphic eta quotients of prime power levels are irreducible.

Motivation & Objective

• To resolve the long-standing open problem of algorithmically determining whether a holomorphic eta quotient is irreducible.
• To address the challenge posed by the lack of known upper bounds on the levels of potential factors of a given eta quotient.
• To establish explicit upper bounds on the minimal level of nontrivial factors of a holomorphic eta quotient in terms of its level $N$.
• To show that for prime power levels $N$, any reducible eta quotient must have a nontrivial factor whose level divides $N$, enabling algorithmic verification.

Proposed method

• The authors analyze the structure of holomorphic eta quotients by examining their levels and the divisibility conditions on potential factor levels.
• They derive an upper bound on the levels of all nontrivial factors of a holomorphic eta quotient $f$ of level $N$, assuming no such factor has level dividing $N$.
• The bound is refined using the weight of the eta quotient, providing a tighter constraint on factor levels.
• The proof leverages properties of modular forms and the transformation behavior of the Dedekind eta function under modular transformations.
• The authors apply Atkin-Lehner involutions and rescaling operations to preserve irreducibility, showing that irreducibility is preserved under these operations for prime power levels.
• They use the theory of modular forms and level-raising techniques to establish that the absence of low-level factors implies boundedness of all factor levels.

Experimental results

Research questions

• RQ1 Can an algorithm be constructed to determine whether a holomorphic eta quotient is irreducible by analyzing its level and factor structure?
• RQ2 Is there an absolute upper bound on the levels of nontrivial factors of a holomorphic eta quotient in terms of its level $N$?
• RQ3 Does every reducible holomorphic eta quotient of prime power level $N$ necessarily possess a nontrivial factor whose level divides $N$?
• RQ4 How does the weight of the eta quotient influence the upper bound on the minimal level of its nontrivial factors?
• RQ5 Are irreducibility properties preserved under rescaling and Atkin-Lehner involutions for eta quotients of prime power levels?

Key findings

• An explicit upper bound on the levels of all nontrivial factors of a holomorphic eta quotient $f$ of level $N$ is established, provided no such factor has level dividing $N$.
• The upper bound on the minimal level of nontrivial factors is expressed explicitly in terms of $N$, with a refined version incorporating the weight of $f$.
• For any reducible holomorphic eta quotient of prime power level $N$, there exists a nontrivial factor whose level divides $N$, which enables algorithmic factorization checks.
• All rescalings by positive integers and all Atkin-Lehner involutions of irreducible holomorphic eta quotients of prime power levels remain irreducible.
• The results imply that irreducibility of eta quotients at prime power levels can be algorithmically determined by checking for factors of level dividing $N$.