Values of harmonic weak Maass forms on Hecke orbits

Abstract

Let q := e(2 pi iz), where z is an element of H. For an even integer k, let f(z) := q(h) Pi(infinity)(m=1)(1-q(m))(c(m)) be a meromorphic modular form of weight k on Gamma(0)(N). For a positive integer m, let T-m, be the mth Hecke operator and D be a divisor of a modular curve with level N. Both subjects, the exponents c(m) of a modular form and the distribution of the points in the support of T-m.D, have been widely investigated. When the level N is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of j-invariant function, identities between the exponents c(m) of a modular form and the points in the support of T-m.D. In this paper, we extend this result to general Gamma(0)(N) in terms of values of harmonic weak Maass forms of weight 0. By the distribution of Hecke points, this applies to obtain an asymptotic behavior of convolutions of sums of divisors of an integer and sums of exponents of a modular form. (C) 2019 Elsevier Inc. All rights reserved.