CONGRUENCES LIKE ATKIN’S FOR THE PARTITION FUNCTION

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(Q^m 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.