TY - JOUR
T1 - Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo ℓ
AU - Choi, D.
N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.
PY - 2009/7
Y1 - 2009/7
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.
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U2 - 10.1090/S0002-9947-09-04708-4
DO - 10.1090/S0002-9947-09-04708-4
M3 - Article
AN - SCOPUS:77950641635
SN - 0002-9947
VL - 361
SP - 3817
EP - 3828
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 7
ER -