TY - JOUR

T1 - Limits of traces of singular moduli

AU - Choi, Dohoon

AU - Lim, Subong

N1 - Funding Information:
Received by the editors December 12, 2018, and, in revised form, April 12, 2019 and April 19, 2019. 2010 Mathematics Subject Classification. Primary 11F37; Secondary 11F30. Key words and phrases. Modular traces, regularized L-functions, Eichler-Shimura cohomology theory. The first author was partially supported by the National Research Foundation of Korea (NRF) grant (NRF-2019R1A2C1007517). The second author was supported by the National Research Foundation of Korea (NRF) grant (NRF-2019R1C1C1009137).

PY - 2020

Y1 - 2020

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

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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 -