### Abstract

Let D be the differential operator defined by D:= [Formula presented] [Formula presented]. This induces a map D^{k+1}:M_{−k} ^{!}(Γ_{0}(N))→M_{k+2} ^{!}(Γ_{0}(N)), where M_{k} ^{!}(Γ_{0}(N)) is the space of weakly holomorphic modular forms of weight k on Γ_{0}(N). The operator D^{k+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 D^{k+1} between dual spaces M_{−k} ^{!}(Γ_{0}(N)) and M_{k+2} ^{!}(Γ_{0}(N)). More precisely, we classify dual pairs (f,D^{k+1}f) under the map D^{k+1} such that f is an eta-quotient and D^{k+1}f 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(d_{i1 }z)^{b1 }⋯η(d_{it }z)^{bt } is less than or equal to 4 and every prime divisor of each d_{i} 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 Γ_{0}(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 language | English |
---|---|

Article number | 106779 |

Journal | Advances in Mathematics |

Volume | 355 |

DOIs | |

Publication status | Published - 2019 Oct 15 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*355*, [106779]. https://doi.org/10.1016/j.aim.2019.106779

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

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 355, 106779. https://doi.org/10.1016/j.aim.2019.106779

}

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 -