Converse conjecture for p-adic annihilation of monstrous moonshine Hauptmoduln

Let $p$ be a prime and let $f$ be a modular function. Say that $f$ is $p$-adically annihilated if the sequence $f\mid U_p^n$ uniformly converges to $0$ in the $p$-adic limit as $n\to\infty$, where

$$\left(\sum a(n)q^n\right)\mid U_p=\sum a(pn)q^n.$$

The Hauptmoduln in the table of the paper are the known $p$-adically annihilated Hauptmoduln among those appearing in monstrous moonshine. Converse conjecture for p-adic annihilation. The converse to the stated annihilation theorem holds: those table-listed Hauptmoduln are the only Hauptmoduln appearing in monstrous moonshine with $p$-adic annihilation. In particular, no Hauptmodul appearing in monstrous moonshine is $p$-adically annihilated for $p\geq13$. This conjecture would give a complete classification of $p$-adic annihilation among the 171 monstrous moonshine Hauptmoduln. The paper presents it as being motivated by computations and heuristics; no resolution is supplied in the given text.