$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$

TL;DR

This paper investigates the p-adic properties of Taylor coefficients of half-integral weight modular forms at CM points, proving they eventually vanish modulo p^m for primes p ≥ 5, extending previous integral weight results.

Contribution

It establishes the vanishing of Taylor coefficients modulo p^m for half-integral weight modular forms on (4), generalizing prior integral weight findings.

Findings

Taylor coefficients vanish modulo p^m for primes p  5

Results extend Larson and Smith's integral weight theorems

Applicable to modular forms at CM points

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 $p\ge5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\mathrm{SL}_2(\mathbb{Z})$.