On $U_p$-congruences for meromorphic modular forms with supersingularity

In this paper, we investigate congruences for meromorphic modular forms $F$ which have a pole at a single point $z$ in the fundamental domain of ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. For a prime $p$ with good supersingular reduction at the elliptic curve corresponding to $z$, we show that there exists a cusp form $f$ such that $F\mathrm{\mid }{U}_p^m\equiv f\mathrm{\mid }{U}_p^m(\mathrm{m}\mathrm{o}\mathrm{d}{p}^{{\kappa }_m})$, where ${\kappa }_m=\alpha m-\beta$ with $\alpha$ only depending on the weight of $F$ and $\beta$ depending on $F$ and $p$ but is independent of $m$. In particular, if the space of cusp forms is trivial, then $F\mathrm{\mid }{U}_p^m\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}{p}^{{\kappa }_m})$ vanishes $p$-adically to a high order. In order to prove these results, we use the fact that $p$ has supersingular reduction to realize $F$ as an overconvergent modular form and then utilize the theory of overconvergent forms to show the congruences.