Wild Stacky Curves and Rings of Mod p Modular Forms

We extend work of Voight and the second author to compute the log canonical ring of a wild stacky curve over a field of characteristic $p > 0$, which allows us to compute rings of mod $p$ modular forms of level $\Gamma_{0}(N)$. Our approach also reveals that in characteristics $2$ and $3$, there are infinitely many levels $N$ for which there are weight $2$ modular forms of level $\Gamma_{0}(N)$ that do not lift to characteristic $0$.