Representations of $SL(2,\mathbb{Z})$ and automorphic forms of singular and critical weight

The study of automorphic forms of singular or critical weight can often be reduced to the study of Jacobi forms of singular or critical weight. These can be interpreted as invariants of Weil representations associated to finite quadratic modules. The latter are the key to the understanding of those representations of $SL(2,\mathbb{Z})$ whose kernel is a congruence subgroup. Finally, these representations are intimately connected to the arithmetic theory of integral quadratic forms. This talk provides an overview of these ideas and their interplay.