### Abstract

In this paper, we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f=∑a(n)q^{n} of weight k on Γ_{1}(N) having infinitely many congruences of the form a(ℓn+β)≡0(modℓ), where ℓ is a prime and β is an integer in (0, 1, ..., ℓ-1). These are similar to congruences for the partition function investigated by Ramanujan. Furthermore, we characterize linear combinations of Shimura theta functions with odd characters in terms of these congruences. As an application of our main theorem, we consider a generalization of Newman's conjecture for weakly holomorphic modular forms on Γ_{0}(N) with real Dirichlet characters. Finally we use these results to study arithmetic properties of colored partitions and generalized Frobenius partitions.

Original language | English |
---|---|

Pages (from-to) | 489-516 |

Number of pages | 28 |

Journal | Advances in Mathematics |

Volume | 294 |

DOIs | |

Publication status | Published - 2016 May 14 |

Externally published | Yes |

### Fingerprint

### Keywords

- Colored partitions
- Congruences for modular forms
- Generalized Frobenius partitions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Congruences involving arithmetic progressions for weakly holomorphic modular forms.** / Choi, Dohoon.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Congruences involving arithmetic progressions for weakly holomorphic modular forms

AU - Choi, Dohoon

PY - 2016/5/14

Y1 - 2016/5/14

N2 - In this paper, we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f=∑a(n)qn of weight k on Γ1(N) having infinitely many congruences of the form a(ℓn+β)≡0(modℓ), where ℓ is a prime and β is an integer in (0, 1, ..., ℓ-1). These are similar to congruences for the partition function investigated by Ramanujan. Furthermore, we characterize linear combinations of Shimura theta functions with odd characters in terms of these congruences. As an application of our main theorem, we consider a generalization of Newman's conjecture for weakly holomorphic modular forms on Γ0(N) with real Dirichlet characters. Finally we use these results to study arithmetic properties of colored partitions and generalized Frobenius partitions.

AB - In this paper, we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f=∑a(n)qn of weight k on Γ1(N) having infinitely many congruences of the form a(ℓn+β)≡0(modℓ), where ℓ is a prime and β is an integer in (0, 1, ..., ℓ-1). These are similar to congruences for the partition function investigated by Ramanujan. Furthermore, we characterize linear combinations of Shimura theta functions with odd characters in terms of these congruences. As an application of our main theorem, we consider a generalization of Newman's conjecture for weakly holomorphic modular forms on Γ0(N) with real Dirichlet characters. Finally we use these results to study arithmetic properties of colored partitions and generalized Frobenius partitions.

KW - Colored partitions

KW - Congruences for modular forms

KW - Generalized Frobenius partitions

UR - http://www.scopus.com/inward/record.url?scp=84960918578&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960918578&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2016.02.032

DO - 10.1016/j.aim.2016.02.032

M3 - Article

AN - SCOPUS:84960918578

VL - 294

SP - 489

EP - 516

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -