### Abstract

Let ℓ be a prime and λ,j ≥ 0 be an integer. Suppose that f(z) = Σ_{n} a(n)q^{n} is a weakly holomorphic modular form of weight λ + ^{1}_{2} and that a(0) ≢ 0 (mod ℓ). We prove that if the coefficients of f(z) are not "well- distributed" modulo ℓ^{j}, 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+ ^{1}_{2} 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 language | English |
---|---|

Pages (from-to) | 3817-3828 |

Number of pages | 12 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2009 Jul 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo ℓ.** / Choi, Dohoon.

Research output: Contribution to journal › Article

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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 -