Limits of traces of singular moduli
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- by Dohoon Choi and Subong Lim PDF
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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 $d$.References
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Additional Information
- Dohoon Choi
- Affiliation: Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Republic of Korea
- MR Author ID: 784974
- 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
- MR Author ID: 893084
- Email: subong@skku.edu
- 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) - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 185-227
- MSC (2010): Primary 11F37; Secondary 11F30
- DOI: https://doi.org/10.1090/tran/7890
- MathSciNet review: 4042872