Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo ℓ

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Let ℓ be a prime and λ,j ≥ 0 be an integer. Suppose that f(z) = Σn a(n)qn is a weakly holomorphic modular form of weight λ + 12 and that a(0) ≢ 0 (mod ℓ). We prove that if the coefficients of f(z) are not "well- distributed" modulo ℓj, then λ = 0 or 1 (mod ℓ-1/2). This implies that, under the additional restriction a(0) ≢ 0 (mod ℓ), the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight λ+ 12 are almost (but not all) divisible by ℓ, then either λ ≡ 0 (mod ℓ-1/2) or λ ≡ 1 (mod ℓ-1/2). We also prove that if λ ≢ 0 and 1 (mod ℓ-1/2), then there does not exist an integer β, 0 ≤ β < ℓ, such that a(ℓn+ β) = 0 (mod ℓ) for every nonnegative integer n. As an application, we study congruences for the values of the overpartition function.

Original languageEnglish
Pages (from-to)3817-3828
Number of pages12
JournalTransactions of the American Mathematical Society
Volume361
Issue number7
DOIs
Publication statusPublished - 2009 Jul

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Fingerprint Dive into the research topics of 'Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo ℓ'. Together they form a unique fingerprint.

  • Cite this