Pairs of eta-quotients with dual weights and their applications

Dohoon Choi, Byungchan Kim, Subong Lim

Research output: Contribution to journalArticle

Abstract

Let D be the differential operator defined by D:= [Formula presented] [Formula presented]. This induces a map Dk+1:M−k !0(N))→Mk+2 !0(N)), where Mk !0(N)) is the space of weakly holomorphic modular forms of weight k on Γ0(N). The operator Dk+1 plays important roles in the theory of Eichler-Shimura cohomology and harmonic weak Maass forms. On the other hand, eta-quotients are fundamental objects in the theory of modular forms and partition functions. In this paper, we show that the structure of eta-quotients is very rarely preserved under the map Dk+1 between dual spaces M−k !0(N)) and Mk+2 !0(N)). More precisely, we classify dual pairs (f,Dk+1f) under the map Dk+1 such that f is an eta-quotient and Dk+1f is a non-zero constant multiple of an eta-quotient. When the levels are square-free, we give the complete classification of such pairs. In general, we find a necessary condition for such pairs: the weight of the primitive eta-quotient f(z)=η(di1 z)b1 ⋯η(dit z)bt is less than or equal to 4 and every prime divisor of each di is less than 11. We also give various applications of these classifications. In particular, we find all eta-quotients of weight 2 and square-free level N such that they are in the Eisenstein space for Γ0(N). To prove our main theorems, we use various combinatorial properties of a Latin square matrix whose rows and columns are exactly divisors of N.

Original languageEnglish
Article number106779
JournalAdvances in Mathematics
Volume355
DOIs
Publication statusPublished - 2019 Oct 15

Fingerprint

Quotient
Square free
Modular Forms
Divisor
Magic square
Dual space
Square matrix
Less than or equal to
Partition Function
Differential operator
Cohomology
Harmonic
Classify
Necessary Conditions
Operator
Theorem

Keywords

  • D-operator
  • Eta-quotient
  • Lambert series
  • Latin matrix

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Pairs of eta-quotients with dual weights and their applications. / Choi, Dohoon; Kim, Byungchan; Lim, Subong.

In: Advances in Mathematics, Vol. 355, 106779, 15.10.2019.

Research output: Contribution to journalArticle

@article{d01cf949915f4344826c1c5bba6b5cdd,
title = "Pairs of eta-quotients with dual weights and their applications",
abstract = "Let D be the differential operator defined by D:= [Formula presented] [Formula presented]. This induces a map Dk+1:M−k !(Γ0(N))→Mk+2 !(Γ0(N)), where Mk !(Γ0(N)) is the space of weakly holomorphic modular forms of weight k on Γ0(N). The operator Dk+1 plays important roles in the theory of Eichler-Shimura cohomology and harmonic weak Maass forms. On the other hand, eta-quotients are fundamental objects in the theory of modular forms and partition functions. In this paper, we show that the structure of eta-quotients is very rarely preserved under the map Dk+1 between dual spaces M−k !(Γ0(N)) and Mk+2 !(Γ0(N)). More precisely, we classify dual pairs (f,Dk+1f) under the map Dk+1 such that f is an eta-quotient and Dk+1f is a non-zero constant multiple of an eta-quotient. When the levels are square-free, we give the complete classification of such pairs. In general, we find a necessary condition for such pairs: the weight of the primitive eta-quotient f(z)=η(di1 z)b1 ⋯η(dit z)bt is less than or equal to 4 and every prime divisor of each di is less than 11. We also give various applications of these classifications. In particular, we find all eta-quotients of weight 2 and square-free level N such that they are in the Eisenstein space for Γ0(N). To prove our main theorems, we use various combinatorial properties of a Latin square matrix whose rows and columns are exactly divisors of N.",
keywords = "D-operator, Eta-quotient, Lambert series, Latin matrix",
author = "Dohoon Choi and Byungchan Kim and Subong Lim",
year = "2019",
month = "10",
day = "15",
doi = "10.1016/j.aim.2019.106779",
language = "English",
volume = "355",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Pairs of eta-quotients with dual weights and their applications

AU - Choi, Dohoon

AU - Kim, Byungchan

AU - Lim, Subong

PY - 2019/10/15

Y1 - 2019/10/15

N2 - Let D be the differential operator defined by D:= [Formula presented] [Formula presented]. This induces a map Dk+1:M−k !(Γ0(N))→Mk+2 !(Γ0(N)), where Mk !(Γ0(N)) is the space of weakly holomorphic modular forms of weight k on Γ0(N). The operator Dk+1 plays important roles in the theory of Eichler-Shimura cohomology and harmonic weak Maass forms. On the other hand, eta-quotients are fundamental objects in the theory of modular forms and partition functions. In this paper, we show that the structure of eta-quotients is very rarely preserved under the map Dk+1 between dual spaces M−k !(Γ0(N)) and Mk+2 !(Γ0(N)). More precisely, we classify dual pairs (f,Dk+1f) under the map Dk+1 such that f is an eta-quotient and Dk+1f is a non-zero constant multiple of an eta-quotient. When the levels are square-free, we give the complete classification of such pairs. In general, we find a necessary condition for such pairs: the weight of the primitive eta-quotient f(z)=η(di1 z)b1 ⋯η(dit z)bt is less than or equal to 4 and every prime divisor of each di is less than 11. We also give various applications of these classifications. In particular, we find all eta-quotients of weight 2 and square-free level N such that they are in the Eisenstein space for Γ0(N). To prove our main theorems, we use various combinatorial properties of a Latin square matrix whose rows and columns are exactly divisors of N.

AB - Let D be the differential operator defined by D:= [Formula presented] [Formula presented]. This induces a map Dk+1:M−k !(Γ0(N))→Mk+2 !(Γ0(N)), where Mk !(Γ0(N)) is the space of weakly holomorphic modular forms of weight k on Γ0(N). The operator Dk+1 plays important roles in the theory of Eichler-Shimura cohomology and harmonic weak Maass forms. On the other hand, eta-quotients are fundamental objects in the theory of modular forms and partition functions. In this paper, we show that the structure of eta-quotients is very rarely preserved under the map Dk+1 between dual spaces M−k !(Γ0(N)) and Mk+2 !(Γ0(N)). More precisely, we classify dual pairs (f,Dk+1f) under the map Dk+1 such that f is an eta-quotient and Dk+1f is a non-zero constant multiple of an eta-quotient. When the levels are square-free, we give the complete classification of such pairs. In general, we find a necessary condition for such pairs: the weight of the primitive eta-quotient f(z)=η(di1 z)b1 ⋯η(dit z)bt is less than or equal to 4 and every prime divisor of each di is less than 11. We also give various applications of these classifications. In particular, we find all eta-quotients of weight 2 and square-free level N such that they are in the Eisenstein space for Γ0(N). To prove our main theorems, we use various combinatorial properties of a Latin square matrix whose rows and columns are exactly divisors of N.

KW - D-operator

KW - Eta-quotient

KW - Lambert series

KW - Latin matrix

UR - http://www.scopus.com/inward/record.url?scp=85071137224&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071137224&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.106779

DO - 10.1016/j.aim.2019.106779

M3 - Article

VL - 355

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 106779

ER -