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Transactions of the American Mathematical Society

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Limits of traces of singular moduli


Authors: Dohoon Choi and Subong Lim
Journal: Trans. Amer. Math. Soc. 373 (2020), 185-227
MSC (2010): Primary 11F37; Secondary 11F30
DOI: https://doi.org/10.1090/tran/7890
Published electronically: August 5, 2019
MathSciNet review: 4042872
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Abstract: Let $ f$ and $ g$ be weakly holomorphic modular functions on $ \Gamma _0(N)$ with the trivial character. For an integer $ d$, let $ \mathrm {Tr}_d(f)$ denote the modular trace of $ f$ of index $ d$. Let $ r$ be a rational number equivalent to $ i\infty $ under the action of $ \Gamma _0(4N)$. In this paper, we prove that when $ z$ goes radially to $ r$, the limit $ Q_{\hat {H}(f)}(r)$ of the sum $ H(f)(z) = \sum _{d>0}\mathrm {Tr}_d(f)e^{2\pi idz}$ is a special value of a regularized twisted $ L$-function defined by $ \mathrm {Tr}_d(f)$ for $ d\leq 0$. It is proved that the regularized $ L$-function is meromorphic on $ \mathbb{C}$ and satisfies a certain functional equation. Finally, under the assumption that $ N$ is square free, we prove that if $ Q_{\hat {H}(f)}(r)=Q_{\hat {H}(g)}(r)$ for all $ r$ equivalent to $ i \infty $ under the action of $ \Gamma _0(4N)$, then $ \mathrm {Tr}_d(f)=\mathrm {Tr}_d(g)$ for all integers $ d$.


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Additional Information

Dohoon Choi
Affiliation: Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Republic of Korea
Email: dohoonchoi@korea.ac.kr

Subong Lim
Affiliation: Department of Mathematics Education, Sungkyunkwan University, 25-2, Sungkyunkwan-ro, Jongno-gu, Seoul 03063, Republic of Korea
Email: subong@skku.edu

DOI: https://doi.org/10.1090/tran/7890
Keywords: Modular traces, regularized $L$-functions, Eichler-Shimura cohomology theory
Received by editor(s): December 12, 2018
Received by editor(s) in revised form: April 12, 2019, and April 19, 2019
Published electronically: August 5, 2019
Additional Notes: 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)
Article copyright: © Copyright 2019 American Mathematical Society