Explainable Ramanujan-type Congruences on Square-Classes of Arithmetic Progressions

Abstract

While examples of Ramanujan-type congruences are amply available via their relation to Hecke operators, it remains unclear which of them should be considered of combinatorial origin and which of them are mere artifacts of the connection with modular forms. Ranks and generalized ranks have been proposed as a tool to discern this question. We formalize this idea as "explainable Ramanujan-type congruences" with reference to Jacobi forms with singularities at torsion points, and then associated with them subspaces in a specific complex representation of the modular group. The resulting representation theoretic perspective allows us to prove that explainable Ramanujan-type congruences occur on square-classes $M \mathbb{Z} + u^2 \beta$ of arithmetic progressions.