Arithmetic properties of the partition function

Ahlgren, Scott; Boylan, Matthew

doi:10.1007/s00222-003-0295-6

Cited by 84 publications

(110 citation statements)

References 21 publications

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“…When the weight is half-integral, the situation is not as clear. The distribution of the coefficients of half-integral weight modular forms in congruence classes has been studied by, among others, Balog, Darmon, and Ono [4], Ono and Skinner [11], [12], Bruinier [5], Bruinier and Ono [6], and Ahlgren and Boylan [1], [2].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The weight of half-integral weight modular forms with few non-vanishing coefficients mod l

Choi

Kilbourn

2007

Acta Arith.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

The weight of half-integral weight modular forms with few non-vanishing coefficients mod l

Choi

Kilbourn

2007

Acta Arith.

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“…Our argument for the proof of Theorem 2 is based on the work of Bruinier and Ono [7] and of Ahlgren and Boylan [1], [3], [4].…”

Section: Lemma 33 Suppose That λ Is a Nonnegative Integer That N Imentioning

confidence: 99%

“…When the weight is half-integral, the situation is not as clear. The distribution of the coefficients of modular forms of half-integral weight in congruence classes has been studied by, among others, Balog, Darmon, and Ono [5], Ono and Skinner [14], [15], Brunier [6], Bruinier and Ono [7], and Ahlgren and Boylan [1], [2]. In this paper, we study congruences for weakly holomorphic modular forms of half-integral weight such that the constant terms of their q-expansions are not zero modulo a prime .…”

Section: Introductionmentioning

confidence: 99%

Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo $\ell $

Choi

2009

Trans. Amer. Math. Soc.

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Abstract. Let be a prime and λ, j ≥ 0 be an integer. Suppose that f (z) = n a(n)q n is a weakly holomorphic modular form of weight λ + 1 2and that a(0) ≡ 0 (mod ). We prove that if the coefficients of f (z) are not "welldistributed" modulo j , thenThis implies that, under the additional restriction a(0) ≡ 0 (mod ), the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight λ+ 1 2 are almost (but not all) divisible by , then either λ ≡ 0 (mod). We also prove that if λ ≡ 0 and 1 (mod −1 2 ), then there does not exist an integer β, 0 ≤ β < , such that a( n + β) ≡ 0 (mod ) for every nonnegative integer n. As an application, we study congruences for the values of the overpartition function.

show abstract

“…Proof of Corollary 1.5. We will show using Proposition 4.2 and arguments of Ono and Ahlgren-Boylan [O2,AB] that if G ,α (z) | T Q 2 ≡ 0 (mod α ) and Q ≡ −1 (mod 24 ), then there is an integer β such that p(Q 3 α n + β ) ≡ 0 (mod α ) for integers n in any of Q − 1 of the residue classes (mod Q). Since …”

Section: Proofs Of Corollariesmentioning

confidence: 99%