Values of harmonic weak Maass forms on Hecke orbits

Abstract

Let q := e 2πiz , where z ∈ H. For an even integer k, let f (z) := q h ∞ m=1 (1 − q m ) c(m) be a meromorphic modular form of weight k on Γ 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 jinvariant 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 Γ 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 behaviour of convolutions of sums of divisors of an integer and sums of exponents of a modular form.

JEON, Beomseok (BJ): Seoul/Republic of Korea

“…The divisor lifting involving the generating function of J n for a weight k meromorphic modular form to a weight 2 meromorphic modular form, as described in (1.4), was generalized by Bringmann et al [3,Theorem 1.3] and Choi, Lee, and Lim [6,Theorem 1.1]. They extended this lifting to meromorphic modular forms of arbitrary level with respect to Niebur-Poincaré harmonic weak Maass functions, resulting in the holomorphic part of the Fourier expansion of a weight 2 polar harmonic weak Maass form.…”

On values of weakly holomorphic modular functions at divisors of meromorphic modular forms

“…These results have been generalized by Bringmann et al [3,Theorem 1.3] and Choi, Lee and Lim [6,Theorem 1.1] to Niebur-Poincaré harmonic weak Maass functions J N,n (τ ) of arbitrary level N . For every point z ∈ H, we define…”

Hecke equivariance of generalized Borcherds products

“…In [10], Bruinier, Kohnen and Ono showed that the logarithmic derivative of a meromorphic modular form f on SL 2 (Z) is expressible in terms of the values of Faber polynomials of j(τ ) at points τ in the divisor of f , which generalizes the denominator formula. These results have been generalized by Bringmann et al [9] and Choi, Lee and Lim [16] to Niebur-Poincaré harmonic weak Maass functions of arbitrary level N . They proved that the logarithmic derivative of a meromorphic modular form for Γ 0 (N ) is explicitly described in terms of the values of Niebur-Poincaré series at its divisors in H. In this paper, we verify that this logarithmic differential operator defined on the multiplicative group of meromorphic modular forms for Γ 0 (N ) with a unitary multiplier system (possibly of infinite order) is also Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator and the Hecke operator on the weight 2 meromorphic modular forms for Γ 0 (N ).…”

“…where F N is the fundamental domain of Γ 0 (N ) and 1/e N,z is the cardinality of Γ 0 (N ) z /{±1} for each z ∈ H, where Γ 0 (N ) z denotes the stabilizer of z in Γ 0 (N ). Bringmann et al [9] and Choi, Lee, and Lim [16] have extended these results to Niebur-Poincaré harmonic weak Maass functions J N,n of arbitrary level N by proving that the logarithmic derivative of a meromorphic modular form for Γ 0 (N ) is described explicitly in terms of the values of J N,n at its divisors in H. In [20, Theorem 2.1], the authors showed that the logarithmic derivative of a meromorphic modular form for Γ 0 (N ) can be expressed explicitly in terms of the values of weakly holomorphic modular functions. Let M !…”