Congruences like Atkin’s for the partition function

Abstract

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