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 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 |
Externally published | Yes |
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics