Congruences for level four cusp forms

Scott Ahlgren, Dohoon Choi, Jeremy Rouse

Research output: Contribution to journalArticle

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 Οo 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=1nλqn2 (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 languageEnglish (US)
Pages (from-to)683-701
Number of pages19
JournalMathematical Research Letters
Volume16
Issue number4
DOIs
StatePublished - Jul 2009

ASJC Scopus subject areas

  • Mathematics(all)

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