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, $2$-adically, to a Hauptmodul on ${\mathrm{\Gamma }}_0(2)$ by using a theorem of Serre. Then, we apply this to show that the image of the character map on the $2$-adic Heisenberg VOA $S_1$ contains the space of $2$-adic overconvergent modular forms $M_2^{†}(1\mathrm{/}2)$ of weight zero.