Congruence relations for r-colored partitions

Abstract

Let ℓ≥5 be prime. For the partition function p(n) and 5≤ℓ≤31, Atkin found a number of examples of primes Q≥5 such that there exist congruences of the form p(ℓQ3n+β)≡0(modℓ). Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every ℓ. In this paper, for a wide range of c∈Fℓ, we prove congruences of the form p(ℓQ3n+β0)≡c⋅p(ℓQn+β1)(modℓ) for infinitely many primes Q. For a positive integer r, let pr(n) be the r-colored partition function. Our methods yield similar congruences for pr(n). In particular, if r is an odd positive integer for which ℓ>5r+19 and 2r+2≢2±1(modℓ), then we show that there are infinitely many congruences of the form pr(ℓQ3n+β)≡0(modℓ). Our methods involve the theory of modular Galois representations.