TY - JOUR

T1 - Special factors of holomorphic eta quotients

AU - Bhattacharya, Soumya

N1 - Funding Information:
This work was done at the Institute of Analysis and Number Theory at Graz University of Technology in Austria and it was supported by the Austrian Science Fund (FWF), Project F-5512 .
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/12/3

Y1 - 2021/12/3

N2 - The levels of the factors of a holomorphic eta quotient f are bounded above with respect to the weight and the level of f. Unfortunately, this bound remains implicit due to the ineffectiveness of Mersmann's finiteness theorem. On the other hand, for checking whether f is irredicble, it is essential to know at least an explicit upper bound for the minimum mf among the levels of the proper factors of f. Recently, such an explicit upper bound has been established. But this bound is far larger than the conjectured minimum. Here, by constructing a special factor of f, we show that the least upper bound for mf is indeed equal to the level of f if the level of f is a prime power. Also, under suitable conditions we generalize the construction of the special factors for holomorphic eta quotients of arbitrary levels.

AB - The levels of the factors of a holomorphic eta quotient f are bounded above with respect to the weight and the level of f. Unfortunately, this bound remains implicit due to the ineffectiveness of Mersmann's finiteness theorem. On the other hand, for checking whether f is irredicble, it is essential to know at least an explicit upper bound for the minimum mf among the levels of the proper factors of f. Recently, such an explicit upper bound has been established. But this bound is far larger than the conjectured minimum. Here, by constructing a special factor of f, we show that the least upper bound for mf is indeed equal to the level of f if the level of f is a prime power. Also, under suitable conditions we generalize the construction of the special factors for holomorphic eta quotients of arbitrary levels.

KW - Dedekind eta function

KW - Eta quotient

KW - Modular form

KW - Special factor

UR - http://www.scopus.com/inward/record.url?scp=85114991073&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.108019

DO - 10.1016/j.aim.2021.108019

M3 - Article

AN - SCOPUS:85114991073

VL - 392

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 108019

ER -