Canonical models of arithmetic $(1; \infty)$ curves

In 1983 Takeuchi showed that up to conjugation there are exactly 4 arithmetic subgroups of $\textrm{PSL}_2 (\mathbb{R})$ with signature $(1; \infty)$. Shinichi Mochizuki gave a purely geometric characterization of the corresponding arithmetic $(1; \infty)$-curves, which also arise naturally in the context of his recent work on inter-universal Teichm\"uller theory. Using Bely\u{\i} maps, we explicitly determine the canonical models of these curves. We also study their arithmetic properties and modular interpretations.