$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on ${\Gamma }_1(4)$

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 ${\theta }_3$ 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 $p\ge 5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on ${\mathrm{SL}}_2(\mathbb{Z})$.

17 pages

References in corpus (2)

• Quasi-Valuations Extending a Valuation
• Periodicities for Taylor coefficients of half-integral weight modular forms