$\mathrm{\ell }$-adic properties and congruences of $\mathrm{\ell }$-regular partition functions

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