ℓ-Adic properties and congruences of ℓ-regular partition functions

We study ℓ -regular partitions by defining a sequence of modular forms of level ℓ and quadratic character which encode their ℓ -adic behavior. We show that this sequence is congruent modulo increasing powers of ℓ to level 1 modular forms of increasing weights. We then prove that certain Z / ℓ m Z -modules generated by our sequence are isomorphic to certain subspaces of level 1 cusp forms of weight independent of the power of ℓ , leading to a uniform bound on the ranks of those modules and consequently to ℓ -adic relations between ℓ -regular partition values.