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 1
Externally publishedYes

Fingerprint

Constant term
Modular Forms
Modulo
Integer
Coefficient
Divisible
Congruence
Non-negative
Restriction
Imply

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

@article{c9b85d8906a94fdbbbd542da666446d4,
title = "Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo ℓ",
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.",
author = "Dohoon Choi",
year = "2009",
month = "7",
day = "1",
doi = "10.1090/S0002-9947-09-04708-4",
language = "English",
volume = "361",
pages = "3817--3828",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "7",

}

TY - JOUR

T1 - Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo ℓ

AU - Choi, Dohoon

PY - 2009/7/1

Y1 - 2009/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77950641635&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950641635&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-09-04708-4

DO - 10.1090/S0002-9947-09-04708-4

M3 - Article

AN - SCOPUS:77950641635

VL - 361

SP - 3817

EP - 3828

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -