Abstract
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.
| 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 |
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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.
In: Advances in Mathematics, Vol. 294, 14.05.2016, p. 489-516.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 -