\(\ell \)-Adic properties and congruences of \(\ell \)-regular partition functions

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