Pairs of eta-quotients with dual weights and their applications

Let D be the differential operator defined by D:= [Formula presented] [Formula presented]. This induces a map D^k+1:M_−k ^!(Γ₀(N))→M_k+2 ^!(Γ₀(N)), where M_k ^!(Γ₀(N)) is the space of weakly holomorphic modular forms of weight k on Γ₀(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 ^!(Γ₀(N)) and M_k+2 ^!(Γ₀(N)). More precisely, we classify dual pairs (f,D^k+1f) under the map D^k+1 such that f is an eta-quotient and D^k+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(d_i1 z)^b₁ ⋯η(d_it z)^b_t 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 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.