Pairs of eta-quotients with dual weights and their applications

Abstract

Let D be the differential operator defined by D := 1/2 pi i d/dz. This induces a map Dk+1 : M--k(!)(Gamma(0)(N)) -> M-k+2(!)(Gamma(0)(N)), where M-k(!)(Gamma(0)(N)) is the space of weakly holomorphic modular forms of weight k on Gamma(0)(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(!) (Gamma(0)(N)) and M-k+2(!)(Gamma(0)(N)). More precisely, we classify dual pairs (f, Dk+1 f) under the map Dk+1 such that f is an eta-quotient and Dk+l 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) = eta(d(i1) z)(b1) ... eta(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 Gamma(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. (C) 2019 Elsevier Inc. All rights reserved.