ℓ-Divisibility of ℓ-regular partition functions

We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function bℓ(n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∏m=1∞(1−qm)ℓ−1.