Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences

Abstract

Let p(n) denote the number of partitions of n. Let A, B ∈ N with A > B and ≥ 5 a prime, such that p(An + B) ≡ 0 (mod ), n ∈ N.Then we will prove that |A and 24B−1 = −1 . This settles an open problem by Scott Ahlgren and Ken Ono. Our proof is based on results by Deligne and Rapoport.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4882 CRISTIAN-SILVIU RADU Many papers have been written on these three congruences and their extensions (already conjectured, and in some cases proved, by Ramanujan) to arbitrary powers of 5, 7, and 11; see the fundamental works of Andrews, Atkin, Berndt, Dyson, Garvan, Kim, Ono, Ramanujan,6, 7,9,11,13,14, 26,23, 24, 25]. According to [3] each of these extensions lies within the class −δ (mod ). We say that a congruence p(An + B) ≡ 0 (mod ), n ∈ Z lies within the class β (mod ) iff {An + B} ⊆ { n + β}. Here for integers x, y we define {xn + y} := {xn + y : n ∈ Z}.The important role that the class −δ (mod ) plays in the theory is illustrated by the work of Kiming and Olsson [15], who proved that if ≥ 5 is prime and p( n + β) ≡ 0 (mod ) for all n, then β ≡ −δ (mod ). Atkin, Newman, O'Brien,8, 10,19] found further congruences modulo m for primes ≤ 31 and small m. Examples by Atkin and Newman in [7] and [19] show that not every congruence lies within the class −δ (mod ). For example when considering = 13, we have

Classification of Congruences for Mock Theta Functions and Weakly Holomorphic Modular Forms

“…We prove analogous results for the mock theta function ω(q) and for a large class of weakly holomorphic modular forms which includes η-quotients. This extends work of Radu [30] in which he proves a conjecture of Ahlgren and Ono for the partition function p(n).…”

“…We now show that M Qℓ,t ≡ 0 (mod ℓ) under the assumption ℓ ∤ (24t − 1). By [30,Lemma 4.6], for each r with 0 ≤ r < ℓ j−1 there exists an integer a r with (a r , 6Qℓ) = 1 such that a 2 r (24t − 1) ≡ 24(t + Qℓr) − 1 (mod Qℓ j ). Therefore t + Qℓr = t A for some A ∈ Γ 0 (2Qℓ j ) as in (3.3), so by Lemma 3.3 we have M Qℓ j ,t+Qℓr ≡ 0 (mod ℓ).…”

“…Therefore j > 0. Lastly, we use [30,Lemma 4.6] as in the proof of Proposition 3.4 to conclude that f Qℓ,t ≡ 0 (mod ℓ) if ℓ ∤ (24t + B).…”

Congruences like Atkin's for the partition function

“…. Radu [Rad13] confirmed a conjecture of the first author and Ono by proving that if there is a congruence p(mn + β) ≡ 0 (mod ℓ) with ℓ ≥ 5 prime, then ℓ | m and 1−24β ℓ ∈ {0, −1}. After this discussion we know that there are many congruences of the form (1.2) with m ≥ 4 and no congruences other then (1.1) with m = 0.…”

Mock theta functions and weakly holomorphic modular forms modulo 2 and 3

“…Finally, we mention that Radu [29] used similar methods to study the behavior of the partition function modulo primes 5. Andersen [4] has obtained similar results in the present context.…”