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

In 1983 Takeuchi showed that up to conjugation there are exactly 4 arithmetic subgroups of ${\text{PSL}}_2(\mathbb{R})$ with signature $(1;\mathrm{\infty })$. Shinichi Mochizuki gave a purely geometric characterization of the corresponding arithmetic $(1;\mathrm{\infty })$-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.