Congruences for Level Four Cusp Forms

In this paper, we study congruences for modular forms of half-integral weight on Γ0(4). Suppose that ` ≥ 5 is prime, that K is a number field, and that v is a prime of K above `. Let Ov denote the ring of v-integral elements of K, and suppose that f(z) = ∑∞ n=1 a(n)q n ∈ Ov[[q]] is a cusp form of weight λ + 1/2 on Γ0(4) in Kohnen’s plus space. We prove that if the coefficients of f are supported on finitely many square classes modulo v and λ+ 1/2 < `(`+ 1 + 1/2), then λ is even and f(z) ≡ a(1) ∞ ∑