PreprintCC BY 4.0arXiv
The structure of irreducible cuspidal eta quotients
Abstract
We show that an ir{\-}reducible cuspidal eta quotient of level~N is uniquely identified by its orders of vanishing at O((log2 N)r−1) cusps, where r denotes the number of distinct prime divisors of N. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely identified only by their orders of vanishing at~∞. We also prove that for all k∈N, there exists a primitive and irreducible cuspidal eta quotient of weight k/2. It follows from the exhaustiveness of Zagier's list of the primitive holomorphic eta quotients of weight~1/2 that the 1st Mersmann level, clM1=12. Here we show that for all k∈N, the k-th Mersmann level is divisible by 22k.
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BibTeX
@misc{bhattacharya2026structure,
title = {{The structure of irreducible cuspidal eta quotients}},
author = {Soumya Bhattacharya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10576},
eprint = {2610.10576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10576},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The structure of irreducible cuspidal eta quotients
AU - Soumya Bhattacharya
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.10576
UR - https://arxiv.org/abs/2610.10576
AB - We show that an ir{\-}reducible cuspidal eta quotient of level~N is uniquely identified by its orders of vanishing at O((log_2 N)^r−1) cusps, where r denotes the number of distinct prime divisors of N. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely identified only by their orders of vanishing at~∞. We also prove that for all k∈N, there exists a primitive and irreducible cuspidal eta quotient of weight k/2. It follows from the exhaustiveness of Zagier's list of the primitive holomorphic eta quotients of weight~1/2 that the 1st Mersmann level, clM_1=12. Here we show that for all k∈N, the k-th Mersmann level is divisible by 2^2k.
ER -