Abstract
For a prime $p \equiv 3 \pmod{4}$ and $m \geq 2$, Romik raised a question about whether the Taylor coefficients around $\sqrt{-1}$ of the classical Jacobi theta function $\theta_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $\Gamma_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p \geq 5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\operatorname{SL}_2(\mathbb{Z})$.
Funding Statement
The authors were supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177). J. Kim was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2020R1I1A1A01074746), and Y. Lee was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2022R1A2C1003203).
Acknowledgments
The authors thank the reviewers for the valuable suggestions.
Citation
Jigu Kim. Yoonjin Lee. "$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$." Taiwanese J. Math. 27 (1) 23 - 38, February, 2023. https://doi.org/10.11650/tjm/220802
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