The Ramanujan Journal

, Volume 48, Issue 2, pp 423–443 | Cite as

Finiteness of irreducible holomorphic eta quotients of a given level

  • Soumya BhattacharyaEmail author


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.


Eta quotient Dedekind eta function Modular form 

Mathematics Subject Classification

Primary 11F20 11F37 11F11 Secondary 11G16 11F12 



I am thankful to Sander Zwegers, who asked during my talk at Cologne whether a Mersmann type finiteness theorem holds if we keep the level of the eta quotients fixed instead of their weight. Corollary 1 is precisely an answer to his question. I would like to thank Don Zagier, Christian Weiß, Danylo Radchenko, Armin Straub, Nadim Rustom, and Christian Kaiser for their comments. I made the computations for the tables using \(\mathtt {PARI/GP}\) [22] and \(\mathtt {Normaliz}\) [21] which I learnt to use from Don and Danylo. I am grateful to them for acquainting me with these very useful computational tools. In particular, Danylo computed \({k_{\max }}(24)\), \({k_{\max }}(28)\) and \({k_{\max }}(30)\) for Table 2. I am grateful to the Max Planck Institute for Mathematics in Bonn and to CIRM : FBK (International Center for Mathematical Research of the Bruno Kessler Foundation) in Trento for providing me with an office space and supporting me with a fellowship during the preparation of this article.


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Authors and Affiliations

  1. 1.CIRM : FBKTrentoItaly
  2. 2.Indian Statistical InstituteKolkataIndia

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