CONGRUENCES LIKE ATKIN’S FOR THE PARTITION FUNCTION

Scott Ahlgren, Patrick B. Allen, Shiang Tang

Research output: Contribution to journalArticlepeer-review

Abstract

Let p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p(Q3 ℓn + β) ≡ 0 (mod ℓ) where ℓ and Q are prime and 5 ≤ ℓ ≤ 31; these lie in two natural families distinguished by the square class of 1−24β (mod ℓ). In recent decades much work has been done to understand congruences of the form p(Qm ℓn + β) ≡ 0 (mod ℓ). It is now known that there are many such congruences when m ≥ 4, that such congruences are scarce (if they exist at all) when m = 1, 2, and that for m = 0 such congruences exist only when ℓ = 5, 7, 11. For congruences like Atkin’s (when m = 3), more examples have been found for 5 ≤ ℓ ≤ 31 but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime ℓ ≥ 5, there are infinitely many congruences like Atkin’s in the first natural family which he discovered and that for at least 17/24 of the primes ℓ there are infinitely many congruences in the second family.

Original languageEnglish (US)
Pages (from-to)1044-1064
Number of pages21
JournalTransactions of the American Mathematical Society Series B
Volume9
Issue number33
DOIs
StatePublished - Dec 8 2022

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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