TY - JOUR

T1 - Values of harmonic weak Maass forms on Hecke orbits

AU - Choi, Dohoon

AU - Lee, Min

AU - Lim, Subong

N1 - Funding Information:
D. Choi was partially supported by the National Research Foundation of Korea (NRF)grant (NRF-2019R1A2C1007517). M. Lee was supported by Royal Society University Research Fellowship ?Automorphic forms, L-functions and trace formulas?. S. Lim was supported by the National Research Foundation of Korea (NRF)grant (NRF-2019R1C1C1009137).

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

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U2 - 10.1016/j.jmaa.2019.04.074

DO - 10.1016/j.jmaa.2019.04.074

M3 - Article

AN - SCOPUS:85065546580

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 -