Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ

Abstract

Let kk be a nonnegative integer. Let KK be a number field and OK{{\mathcal{O}}}_{K} be the ring of integers of KK. Let ℓ≥5\ell \ge 5 be a prime and vv be a prime ideal of OK{{\mathcal{O}}}_{K} over ℓ\ell . Let ff be a modular form of weight k+12k+\frac{1}{2} on Γ0{\Gamma }_{0}(4) such that its Fourier coefficients are in OK{{\mathcal{O}}}_{K}. In this article, we study sufficient conditions that if ff has the form f(z)≡∑n=1∞∑i=1taf(sin2)qsin2(modv)f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers si{s}_{i}, then ff is congruent to a linear combination of iterated derivatives of a single theta function modulo vv.

Keywords

• fourier coefficients of modular forms
• galois representations
• modular forms of half-integral weight
• theta functions
• 11f33
• 11f80