p-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on Γ₁(4)

For a prime p ≡ 3 (mod 4) and m ≥ 2, Romik raised a question about whether the Taylor coefficients around^√−1 of the classical Jacobi theta function θ₃ eventually vanish modulo p^m. This question can be extended to a class of modular forms of half-integral weight on Γ₁(4) and CM points; in this paper, we prove an affirmative answer to it for primes p ≥ 5. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on SL₂(Z).