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

Abstract

In 1983 Takeuchi showed that up to conjugation there are exactly 4 arithmetic subgroups of ${\text{PSL}}_2(\mathbb{R})$ 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 Teichmüller theory. Using Bely\uı maps, we explicitly determine the canonical models of these curves. We also study their arithmetic properties and modular interpretations.

arXiv comment19 pages; updated abstract to be more informative