An Arithmetic Formula for the Partition Function

Although there is a vast literature on the properties of p(n), typically motivated by work of Ramanujan, some of the simplest questions remain open. For example, little is known about p(n) modulo 2 and 3. Most results concerning the congruence properties of p(n) have been proved using properties of (1.1). Theorems typically depend on q-series identities, the theory of modular equations, or the theory of Hecke operators acting on p-adic modular forms. Here we propose a new approach, one based on the properties of algebraic numbers associated to CM points. Before we discuss the partition function, we begin by considering the q-series