Congruences for level four cusp forms

In this paper, we study congruences for modular forms of half-integral weight on Γ(4). Suppose that l ≤ 5 is prime, that K is a number field, and that v is a prime of K above l. Let Ο_o denote the ring of v-integral elements of K, and suppose that f(z) = Σ ^∞_n=1 a(n)qⁿ ε Ο _v[[q]] is a cusp form of weight λ + 1/2 on Γ 0(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 < l(l + 1 + 1/2), then λ is even and f(z) = a(1)Σ^∞ _n=1n^λqⁿ² (mod v). This result is a precise analogue of a characteristic zero theorem of Vignéras [22]. As an application, we study divisibility properties of the algebraic parts of the central critical values of modular L-functions.