### Abstract

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)q^{n} 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=1}n^{λ}q^{n2} (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.

Original language | English |
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Pages (from-to) | 683-701 |

Number of pages | 19 |

Journal | Mathematical Research Letters |

Volume | 16 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 Jul |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematical Research Letters*,

*16*(4), 683-701. https://doi.org/10.4310/MRL.2009.v16.n4.a10