An improved lower bound on the image of the 2-adic character map for the Heisenberg algebra via modular linear differential equations

We describe families of MLDEs whose solutions are modular forms of level one that converge, [Formula: see text]-adically, to a Hauptmodul on [Formula: see text] by using a theorem of Serre. Then, we apply this to show that the image of the character map on the [Formula: see text]-adic Heisenberg VOA [Formula: see text] contains the space of [Formula: see text]-adic overconvergent modular forms [Formula: see text] of weight zero.