### Abstract

Let f and g be weakly holomorphic modular functions on Γ0(N) with the trivial character. For an integer d, let Trd(f) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ0(4N). In this paper, we prove that when z goes radially to r, the limit QH (f)(r) of the sum H(f)(z) =Σd>0 Trd(f)e2πidz is a special value of a regularized twisted L-function defined by Trd(f) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QH(f)(r) = QH (g)(r) for all r equivalent to i∞ under the action of Γ0(4N), then Trd(f) = Trd(g) for all integers d.

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

Pages (from-to) | 185-227 |

Number of pages | 43 |

Journal | Transactions of the American Mathematical Society |

Volume | 373 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2020 Jan 1 |

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### Keywords

- Eichler-Shimura cohomology theory
- Modular traces
- Regularized L-functions

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*373*(1), 185-227. https://doi.org/10.1090/tran/7890

**Limits of traces of singular moduli.** / Choi, Dohoon; Lim, Subong.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 373, no. 1, pp. 185-227. https://doi.org/10.1090/tran/7890

}

TY - JOUR

T1 - Limits of traces of singular moduli

AU - Choi, Dohoon

AU - Lim, Subong

PY - 2020/1/1

Y1 - 2020/1/1

N2 - Let f and g be weakly holomorphic modular functions on Γ0(N) with the trivial character. For an integer d, let Trd(f) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ0(4N). In this paper, we prove that when z goes radially to r, the limit QH (f)(r) of the sum H(f)(z) =Σd>0 Trd(f)e2πidz is a special value of a regularized twisted L-function defined by Trd(f) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QH(f)(r) = QH (g)(r) for all r equivalent to i∞ under the action of Γ0(4N), then Trd(f) = Trd(g) for all integers d.

AB - Let f and g be weakly holomorphic modular functions on Γ0(N) with the trivial character. For an integer d, let Trd(f) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ0(4N). In this paper, we prove that when z goes radially to r, the limit QH (f)(r) of the sum H(f)(z) =Σd>0 Trd(f)e2πidz is a special value of a regularized twisted L-function defined by Trd(f) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QH(f)(r) = QH (g)(r) for all r equivalent to i∞ under the action of Γ0(4N), then Trd(f) = Trd(g) for all integers d.

KW - Eichler-Shimura cohomology theory

KW - Modular traces

KW - Regularized L-functions

UR - http://www.scopus.com/inward/record.url?scp=85077387338&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85077387338&partnerID=8YFLogxK

U2 - 10.1090/tran/7890

DO - 10.1090/tran/7890

M3 - Article

AN - SCOPUS:85077387338

VL - 373

SP - 185

EP - 227

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -