turkmath.org

In this talk, we discuss a framework for studying  $\ell$-regular partitions by defining a sequence of modular forms of level $\ell$ and quadratic character which encode the $\ell$-adic behavior of the so-called $\ell$-regular partitoins. We show that this sequence is congruent modulo increasing powers of $\ell$ to level 1 modular forms of increasing weights. We then prove that certain modules generated by our sequence are isomorphic to certain subspaces of level 1 cusp forms of weight independent of the power of $\ell$, leading to a uniform bound on the ranks of those modules and consequently to $\ell$-adic relations between $\ell$-regular partition values. This generalizes earlier work of Folsom, Kent and Ono on the partition function, where the relevant forms had no nebentypus, and is joint work with Mostafa Ghazy.