TY - JOUR

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

AU - Choi, Dohoon

AU - Kim, Byungchan

AU - Lim, Subong

N1 - Funding Information:
Dohoon Choi was partially supported by the National Research Foundation of Korea (NRF) grant (NRF-2019R1A2C1007517). Byungchan Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1A09917344). Subong Lim was supported by the National Research Foundation of Korea (NRF) grant (NRF-2019R1C1C1009137). We are grateful to the anonymous referees for their very careful readings and valuable comments, which improved the exposition of the paper a lot.

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

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U2 - 10.1016/j.aim.2019.106779

DO - 10.1016/j.aim.2019.106779

M3 - Article

AN - SCOPUS:85071137224

VL - 355

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 106779

ER -