On p-adic modularity in the p-adic Heisenberg algebra

Abstract

We establish existence theorems for the image of the normalized character map of the p-adic Heisenberg algebra S taking values in the algebra of Serre p-adic modular forms $M_p$. In particular, we describe the construction of an analytic family of states in S whose character values are the well-known Λ-adic family of p-adic Eisenstein series of level one built from classical Eisenstein series. This extends previous work treating a specialization at weight 2, and illustrates that the image of the character map contains nonzero p-adic modular forms of every p-adic weight. In a different direction, we prove that for $p=2$ the image of the rescaled character map contains every overconvergent 2-adic modular form of weight zero and tame level one; in particular, it contains the polynomial algebra $Q_2\left[j^{-1}\right]$. For general primes p, we study the square-bracket formalism for S and develop the idea that although states in S do not generally have a conformal weight, they can acquire a p-adic weight in the sense of Serre.