Congruences for level four cusp forms

In this paper, we study congruences for modular forms of half-integral weight on $\Gamma_0(4)$. Suppose that $\ell\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime of $K$ above $\ell$. Let $\mathcal{O}_{v}$ denote the ring of $v$-integral elements of $K$, and suppose that $f(z)=\sum_{n=1}^{\infty}a(n)q^n \in \mathcal{O}_v[[q]]$ is a cusp form of weight $\lambda+1/2$ on $\Gamma_0(4)$ in Kohnen’s plus space. We prove that if the coefficients of $f$ are supported on finitely many square classes modulo $v$ and $\lambda + 1/2 < \ell(\ell + 1 + 1/2)$, then $\lambda$ is even and \[f(z)\equiv a(1) \sum_{n=1}^{\infty}n^{\lambda}q^{n^2} \pmod{v}. \] This result is a precise analogue of a characteristic zero theorem of Vignéras \cite{V}. As an application, we study divisibility properties of the algebraic parts of the central critical values of modular $L$-functions.

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Published 1 January 2009