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arXiv · 2109.08778

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $Γ_1(4)$

Jigu Kim·Yoonjin Lee
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Abstract

For a prime $p\equiv 3$ $(\text{mod }4)$ and $m\ge 2$, Romik raised a question about whether the Taylor coefficients around $\sqrt{-1}$ of the classical Jacobi theta function $θ_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $Γ_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p\ge5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\mathrm{SL}_2(\mathbb{Z})$.

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Jigu Kim, Yoonjin Lee. 2022-09-06. $p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $Γ_1(4)$. https://doi.org/10.11650/tjm%2F220802

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