Congruences involving arithmetic progressions for weakly holomorphic modular forms

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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 languageEnglish
Pages (from-to)489-516
Number of pages28
JournalAdvances in Mathematics
Volume294
DOIs
Publication statusPublished - 2016 May 14
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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