Geodesic continued fraction for Shimura curves and its periodicity: the case of $(2,3,7)$-triangle group

TL;DR

This paper develops a continued fraction expansion for real numbers associated with the Shimura curve from the (2,3,7)-triangle group, proving a periodicity theorem linked to units in quadratic extensions.

Contribution

It introduces a new continued fraction expansion based on geodesic coding on the Shimura curve and establishes a Lagrange type periodicity theorem for this expansion.

Findings

Proves periodicity of the continued fraction expansion for certain quadratic fields.

Connects the expansion's periodicity to fundamental units of quadratic extensions.

Discusses convergence properties of the new continued fractions.

Abstract

In this paper we study the geodesic continued fraction in the case of the Shimura curve coming from the $(2,3,7)$-triangle group. We construct a certain continued fraction expansion of real numbers using the so-called coding of the geodesics on the Shimura curve, and prove the Lagrange type periodicity theorem for the expansion which captures the fundamental relative units of quadratic extensions of $\mathbb{Q}(\cos(2\pi/7))$ with rank one relative unit groups. We also discuss the convergence of these continued fractions.