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

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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