### 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 |
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Article number | 106779 |

Journal | Advances in Mathematics |

Volume | 355 |

DOIs | |

Publication status | Published - 2019 Oct 15 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

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