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

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 $\Lambda$-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 $\mathbf{Q}_2[j^{-1}]$. 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.