Modular forms of half-integral weight on Gamma(0)(4) with few nonvanishing coefficients modulo l

Abstract

Let k be a nonnegative integer. Let K be a number field and O-K be the ring of integers of K. Let l >= 5 be a prime and v be a prime ideal of O-K over l. Let f be a modular form of weight k + 1/2 on Gamma(0)(4) such that its Fourier coefficients are in O-K. In this article, we study sufficient conditions that if f has the form f(z) Xi Sigma(infinity)(n=1)Sigma(t)(j=1) af(s(i)n(2))q(i)(s)n2 (mod v) with square-free integers s(i), then f is congruent to a linear combination of iterated derivatives of a single theta function modulo v.