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arXiv · 2211.13910

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

Hohto Bekki
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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.

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Hohto Bekki. 2022-11-25. Geodesic continued fraction for Shimura curves and its periodicity: the case of $(2,3,7)$-triangle group. https://arxiv.org/abs/2211.13910

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