### 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 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 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 behavior of convolutions of sums of divisors of an integer and sums of exponents of a modular form.

Original language | English |
---|---|

Pages (from-to) | 1046-1062 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 477 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2019 Sep 15 |

### Fingerprint

### Keywords

- Distribution
- Harmonic weak Maass forms
- Hecke orbits

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*477*(2), 1046-1062. https://doi.org/10.1016/j.jmaa.2019.04.074

**Values of harmonic weak Maass forms on Hecke orbits.** / Choi, Dohoon; Lee, Min; Lim, Subong.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 477, no. 2, pp. 1046-1062. https://doi.org/10.1016/j.jmaa.2019.04.074

}

TY - JOUR

T1 - Values of harmonic weak Maass forms on Hecke orbits

AU - Choi, Dohoon

AU - Lee, Min

AU - Lim, Subong

PY - 2019/9/15

Y1 - 2019/9/15

N2 - Let q:=e2πiz, where z∈H. For an even integer k, let f(z):=qh∏m=1 ∞(1−qm)c(m) be a meromorphic modular form of weight k on Γ0(N). For a positive integer m, let Tm 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 Tm.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 Tm.D. In this paper, we extend this result to general Γ0(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.

AB - Let q:=e2πiz, where z∈H. For an even integer k, let f(z):=qh∏m=1 ∞(1−qm)c(m) be a meromorphic modular form of weight k on Γ0(N). For a positive integer m, let Tm 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 Tm.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 Tm.D. In this paper, we extend this result to general Γ0(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.

KW - Distribution

KW - Harmonic weak Maass forms

KW - Hecke orbits

UR - http://www.scopus.com/inward/record.url?scp=85065546580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065546580&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2019.04.074

DO - 10.1016/j.jmaa.2019.04.074

M3 - Article

VL - 477

SP - 1046

EP - 1062

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -